### The Mizar article:

### On the Subcontinua of a Real Line

**by****Adam Grabowski**

- Received June 12, 2003
Copyright (c) 2003 Association of Mizar Users

- MML identifier: BORSUK_5
- [ MML identifier index ]

environ vocabulary BOOLE, ARYTM_1, PRE_TOPC, COMPTS_1, SEQ_1, RELAT_2, BORSUK_1, SUBSET_1, RCOMP_1, TOPMETR, CONNSP_1, ARYTM_3, SQUARE_1, RELAT_1, ORDINAL2, LATTICES, SEQ_2, INTEGRA1, FUNCT_1, TREAL_1, ARYTM, LIMFUNC1, BORSUK_2, PCOMPS_1, GRAPH_1, INCPROJ, CARD_1, SETFAM_1, RAT_1, METRIC_1, FINSET_1, ABSVALUE, IRRAT_1, POWER, INT_1, BORSUK_5, XREAL_0; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, ENUMSET1, SQUARE_1, RELAT_1, FUNCT_1, FUNCT_2, STRUCT_0, INT_1, PRE_TOPC, COMPTS_1, RCOMP_1, SEQ_4, TREAL_1, RCOMP_2, CONNSP_1, BORSUK_2, CARD_1, TOPMETR, PSCOMP_1, INTEGRA1, LIMFUNC1, IRRAT_1, FINSET_1, ABSVALUE, INCPROJ, RAT_1, METRIC_1, PCOMPS_1, NECKLACE; constructors TOPS_2, CONNSP_1, RCOMP_2, PSCOMP_1, INTEGRA1, BORSUK_3, COMPTS_1, YELLOW_8, PARTFUN1, LIMFUNC1, TREAL_1, POWER, PROB_1, BORSUK_2, ENUMSET1, NAT_1, INCPROJ, URYSOHN1, RAT_1, NECKLACE, MEMBERED; clusters XREAL_0, RELSET_1, TOPREAL6, STRUCT_0, PRE_TOPC, BORSUK_2, INTEGRA1, YELLOW13, TOPS_1, JORDAN6, BORSUK_4, FINSET_1, RAT_1, METRIC_1, INT_1, MEMBERED, ZFMISC_1; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; definitions XBOOLE_0, TARSKI; theorems TOPS_2, BORSUK_1, RCOMP_2, PRE_TOPC, COMPTS_1, TARSKI, TOPREAL5, TOPMETR, RCOMP_1, CONNSP_1, ZFMISC_1, JORDAN5A, AXIOMS, XREAL_0, SQUARE_1, REAL_1, TOPREAL6, TREAL_1, XBOOLE_0, XBOOLE_1, JORDAN6, SEQ_4, JGRAPH_5, INTEGRA1, FUNCT_2, FUNCT_1, TOPS_1, XCMPLX_0, BORSUK_4, LIMFUNC1, PROB_1, BORSUK_2, MEASURE1, ENUMSET1, RAT_1, METRIC_1, HAUSDORF, ABSVALUE, XCMPLX_1, TOPREAL3, INCPROJ, CARD_2, NECKLACE, REAL_2, INT_1, YELLOW_8, FCONT_3, SETWISEO, IRRAT_1; begin :: Preliminaries canceled; theorem Th2: for A, B, a being set st A c= B & B c= A \/ {a} holds A \/ {a} = B or A = B proof let A, B, a be set; assume A1: A c= B & B c= A \/ {a}; assume A \/ {a} <> B & A <> B; then A2: not A \/ {a} c= B & not B c= A by A1,XBOOLE_0:def 10; not a in B proof assume a in B; then {a} c= B by ZFMISC_1:37; hence thesis by A1,A2,XBOOLE_1:8; end; hence thesis by A1,A2,SETWISEO:5; end; theorem for x1, x2, x3, x4, x5, x6 being set holds { x1, x2, x3, x4, x5, x6 } = { x1, x3, x6 } \/ { x2, x4, x5 } proof let x1, x2, x3, x4, x5, x6 be set; thus { x1, x2, x3, x4, x5, x6 } c= { x1, x3, x6 } \/ { x2, x4, x5 } proof let x be set; assume x in { x1, x2, x3, x4, x5, x6 }; then x = x1 or x = x2 or x = x3 or x = x4 or x = x5 or x = x6 by ENUMSET1:28; then x in { x1, x3, x6 } or x in { x2, x4, x5 } by ENUMSET1:14; hence thesis by XBOOLE_0:def 2; end; let x be set; assume x in { x1, x3, x6 } \/ { x2, x4, x5 }; then x in { x1, x3, x6 } or x in { x2, x4, x5 } by XBOOLE_0:def 2; then x = x1 or x = x3 or x = x6 or x = x2 or x = x4 or x = x5 by ENUMSET1:13; hence thesis by ENUMSET1:29; end; reserve x1, x2, x3, x4, x5, x6, x7 for set; definition let x1, x2, x3, x4, x5, x6 be set; pred x1, x2, x3, x4, x5, x6 are_mutually_different means :Def1: x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x4 <> x5 & x4 <> x6 & x5 <> x6; end; definition let x1, x2, x3, x4, x5, x6, x7 be set; pred x1, x2, x3, x4, x5, x6, x7 are_mutually_different means :Def2: x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x1 <> x7 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x2 <> x7 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x3 <> x7 & x4 <> x5 & x4 <> x6 & x4 <> x7 & x5 <> x6 & x5 <> x7 & x6 <> x7; end; theorem Th4: for x1, x2, x3, x4, x5, x6 being set st x1, x2, x3, x4, x5, x6 are_mutually_different holds card {x1, x2, x3, x4, x5, x6} = 6 proof let x1, x2, x3, x4, x5, x6 be set; assume x1, x2, x3, x4, x5, x6 are_mutually_different; then A1: x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x4 <> x5 & x4 <> x6 & x5 <> x6 by Def1; then x1, x2, x3, x4, x5 are_mutually_different by NECKLACE:def 1; then A2: card {x1,x2,x3,x4,x5} = 5 by NECKLACE:1; not x6 in {x1,x2,x3,x4,x5} & {x1,x2,x3,x4,x5} is finite & {x1,x2,x3,x4,x5,x6} = {x1,x2,x3,x4,x5} \/ {x6} by A1,ENUMSET1:23,55; hence card {x1,x2,x3,x4,x5,x6} = 5+1 by A2,CARD_2:54 .= 6; end; theorem for x1, x2, x3, x4, x5, x6, x7 being set st x1, x2, x3, x4, x5, x6, x7 are_mutually_different holds card {x1, x2, x3, x4, x5, x6, x7} = 7 proof let x1, x2, x3, x4, x5, x6, x7 be set; assume x1, x2, x3, x4, x5, x6, x7 are_mutually_different; then A1: x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x1 <> x7 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x2 <> x7 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x3 <> x7 & x4 <> x5 & x4 <> x6 & x4 <> x7 & x5 <> x6 & x5 <> x7 & x6 <> x7 by Def2; then x1, x2, x3, x4, x5, x6 are_mutually_different by Def1; then A2: card {x1,x2,x3,x4,x5,x6} = 6 by Th4; not x7 in {x1,x2,x3,x4,x5,x6} & {x1,x2,x3,x4,x5,x6} is finite & {x1,x2,x3,x4,x5,x6,x7} = {x1,x2,x3,x4,x5,x6} \/ {x7} by A1,ENUMSET1:28,61; hence card {x1,x2,x3,x4,x5,x6,x7} = 6+1 by A2,CARD_2:54 .= 7; end; theorem Th6: { x1, x2, x3 } misses { x4, x5, x6 } implies x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6 proof assume A1: { x1, x2, x3 } misses { x4, x5, x6 }; assume x1 = x4 or x1 = x5 or x1 = x6 or x2 = x4 or x2 = x5 or x2 = x6 or x3 = x4 or x3 = x5 or x3 = x6; then A2: x4 in { x1, x2, x3 } or x5 in { x1, x2, x3 } or x6 in { x1, x2, x3 } by ENUMSET1:14; x4 in { x4, x5, x6 } & x5 in { x4, x5, x6 } & x6 in { x4, x5, x6 } by ENUMSET1:14; hence thesis by A1,A2,XBOOLE_0:3; end; theorem x1, x2, x3 are_mutually_different & x4, x5, x6 are_mutually_different & { x1, x2, x3 } misses { x4, x5, x6 } implies x1, x2, x3, x4, x5, x6 are_mutually_different proof assume that A1: x1, x2, x3 are_mutually_different and A2: x4, x5, x6 are_mutually_different and A3: { x1, x2, x3 } misses { x4, x5, x6 }; A4: x1 <> x2 & x2 <> x3 & x1 <> x3 by A1,INCPROJ:def 5; A5: x4 <> x5 & x5 <> x6 & x4 <> x6 by A2,INCPROJ:def 5; x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6 by A3,Th6; hence thesis by A4,A5,Def1; end; theorem x1, x2, x3, x4, x5, x6 are_mutually_different & { x1, x2, x3, x4, x5, x6 } misses { x7 } implies x1, x2, x3, x4, x5, x6, x7 are_mutually_different proof assume that A1: x1, x2, x3, x4, x5, x6 are_mutually_different and A2: { x1, x2, x3, x4, x5, x6 } misses { x7 }; A3: x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x4 <> x5 & x4 <> x6 & x5 <> x6 by A1,Def1; not x7 in { x1, x2, x3, x4, x5, x6 } by A2,ZFMISC_1:54; then x7 <> x1 & x7 <> x2 & x7 <> x3 & x7 <> x4 & x7 <> x5 & x7 <> x6 by ENUMSET1:29; hence thesis by A3,Def2; end; theorem x1, x2, x3, x4, x5, x6, x7 are_mutually_different implies x7, x1, x2, x3, x4, x5, x6 are_mutually_different proof assume x1, x2, x3, x4, x5, x6, x7 are_mutually_different; then x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x1 <> x7 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x2 <> x7 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x3 <> x7 & x4 <> x5 & x4 <> x6 & x4 <> x7 & x5 <> x6 & x5 <> x7 & x6 <> x7 by Def2; hence thesis by Def2; end; theorem x1, x2, x3, x4, x5, x6, x7 are_mutually_different implies x1, x2, x5, x3, x6, x7, x4 are_mutually_different proof assume x1, x2, x3, x4, x5, x6, x7 are_mutually_different; then x1 <> x2 & x1 <> x3 & x1 <> x4 & x1 <> x5 & x1 <> x6 & x1 <> x7 & x2 <> x3 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x2 <> x7 & x3 <> x4 & x3 <> x5 & x3 <> x6 & x3 <> x7 & x4 <> x5 & x4 <> x6 & x4 <> x7 & x5 <> x6 & x5 <> x7 & x6 <> x7 by Def2; hence thesis by Def2; end; theorem Th11: for T being non empty TopSpace, a, b being Point of T st (ex f being map of I[01], T st f is continuous & f.0 = a & f.1 = b) holds ex g being map of I[01], T st g is continuous & g.0 = b & g.1 = a proof let T be non empty TopSpace, a, b be Point of T; given P being map of I[01], T such that A1: P is continuous & P.0 = a & P.1 = b; set e = L[01]((0,1)(#),(#)(0,1)); A2: e.0 = e.(#)(0,1) by TREAL_1:def 1 .= (0,1)(#) by TREAL_1:12 .= 1 by TREAL_1:def 2; A3: e.1 = e.(0,1)(#) by TREAL_1:def 2 .= (#)(0,1) by TREAL_1:12 .= 0 by TREAL_1:def 1; set f = P * e; f is Function of the carrier of Closed-Interval-TSpace (0,1), the carrier of T by FUNCT_2:19,TOPMETR:27; then reconsider f as map of I[01], T by TOPMETR:27; 0 in [. 0,1 .] by TOPREAL5:1; then 0 in the carrier of Closed-Interval-TSpace (0,1) by TOPMETR:25; then A4: 0 in dom e by FUNCT_2:def 1; 1 in [. 0,1 .] by TOPREAL5:1; then 1 in the carrier of Closed-Interval-TSpace (0,1) by TOPMETR:25; then A5: 1 in dom e by FUNCT_2:def 1; take f; e is continuous map of Closed-Interval-TSpace(0,1),Closed-Interval-TSpace(0,1) by TREAL_1:11; hence f is continuous by A1,TOPMETR:27,TOPS_2:58; thus thesis by A1,A2,A3,A4,A5,FUNCT_1:23; end; Lm1: R^1 is arcwise_connected proof for a, b being Point of R^1 ex f being map of I[01], R^1 st f is continuous & f.0 = a & f.1 = b proof let a, b be Point of R^1; per cases; suppose A1: a <= b; reconsider B = [. a, b .] as Subset of R^1 by TOPMETR:24; reconsider B as non empty Subset of R^1 by A1,TOPREAL5:1; A2: Closed-Interval-TSpace(a,b) = R^1|B by A1,TOPMETR:26; then A3: L[01]((#)(a,b),(a,b)(#)) is continuous map of I[01], R^1|B by A1,TOPMETR:27,TREAL_1:11; the carrier of R^1|B c= the carrier of R^1 by BORSUK_1:35; then reconsider g = L[01]((#)(a,b),(a,b)(#)) as Function of the carrier of I[01], the carrier of R^1 by A2,FUNCT_2:9,TOPMETR:27; reconsider g as map of I[01], R^1; take g; thus g is continuous by A3,TOPMETR:7; 0 = (#)(0,1) by TREAL_1:def 1; hence g.0 = (#)(a,b) by A1,TREAL_1:12 .= a by A1,TREAL_1:def 1; 1 = (0,1)(#) by TREAL_1:def 2; hence g.1 = (a,b)(#) by A1,TREAL_1:12 .= b by A1,TREAL_1:def 2; suppose A4: b <= a; reconsider B = [. b, a .] as Subset of R^1 by TOPMETR:24; b in B by A4,TOPREAL5:1; then reconsider B = [. b, a .] as non empty Subset of R^1; A5: Closed-Interval-TSpace(b,a) = R^1|B by A4,TOPMETR:26; then A6: L[01]((#)(b,a),(b,a)(#)) is continuous map of I[01], R^1|B by A4,TOPMETR:27,TREAL_1:11; the carrier of R^1|B c= the carrier of R^1 by BORSUK_1:35; then reconsider g = L[01]((#)(b,a),(b,a)(#)) as Function of the carrier of I[01], the carrier of R^1 by A5,FUNCT_2:9,TOPMETR:27; reconsider g as map of I[01], R^1; A7: g is continuous by A6,TOPMETR:7; 0 = (#)(0,1) by TREAL_1:def 1; then A8: g.0 = (#)(b,a) by A4,TREAL_1:12 .= b by A4,TREAL_1:def 1; 1 = (0,1)(#) by TREAL_1:def 2; then A9: g.1 = (b,a)(#) by A4,TREAL_1:12 .= a by A4,TREAL_1:def 2; then reconsider P = g as Path of b, a by A7,A8,BORSUK_2:def 1; take h = -P; ex t being map of I[01], R^1 st t is continuous & t.0 = a & t.1 = b by A7,A8,A9,Th11; hence thesis by BORSUK_2:def 1; end; hence thesis by BORSUK_2:def 2; end; definition cluster R^1 -> arcwise_connected; coherence by Lm1; end; definition cluster connected non empty TopSpace; existence proof take R^1; thus thesis; end; end; begin :: Intervals theorem for A being Subset of REAL holds A is Subset of R^1 by TOPMETR:24; theorem Th13: [#]R^1 = REAL by PRE_TOPC:def 3,TOPMETR:24; Lm2: for A being Subset of REAL, B being Subset of R^1 st A = B holds Cl A = Cl B by TOPREAL6:80; definition let a be real number; redefine func left_closed_halfline a; :: LIMFUNC1 synonym ].-infty, a .]; redefine func left_open_halfline a; synonym ].-infty, a .[; redefine func right_closed_halfline a; synonym [. a,+infty.[; redefine func right_open_halfline a; synonym ]. a,+infty.[; end; theorem Th14: for a, b being real number holds a in ]. b,+infty.[ iff a > b proof let a, b be real number; A1: a is Element of REAL by XREAL_0:def 1; hereby assume a in ]. b,+infty.[; then a in { g where g is Element of REAL : b < g } by LIMFUNC1:def 3; then consider h being Element of REAL such that A2: h = a & b < h; thus b < a by A2; end; assume b < a; then a in { g where g is Element of REAL : b < g } by A1; hence thesis by LIMFUNC1:def 3; end; theorem Th15: for a, b being real number holds a in [. b,+infty.[ iff a >= b proof let a, b be real number; A1: a is Element of REAL by XREAL_0:def 1; hereby assume a in [. b,+infty.[; then a in { g where g is Element of REAL : b <= g } by LIMFUNC1:def 2; then consider h being Element of REAL such that A2: h = a & b <= h; thus b <= a by A2; end; assume b <= a; then a in { g where g is Element of REAL : b <= g } by A1; hence thesis by LIMFUNC1:def 2; end; theorem Th16: for a, b being real number holds a in ].-infty, b .] iff a <= b proof let a, b be real number; A1: a is Element of REAL by XREAL_0:def 1; hereby assume a in ].-infty,b .]; then a in { g where g is Element of REAL : g <= b } by LIMFUNC1:def 1; then consider h being Element of REAL such that A2: h = a & b >= h; thus b >= a by A2; end; assume b >= a; then a in { g where g is Element of REAL : b >= g } by A1; hence thesis by LIMFUNC1:def 1; end; theorem Th17: for a, b being real number holds a in ].-infty, b .[ iff a < b proof let a, b be real number; A1: a is Element of REAL by XREAL_0:def 1; hereby assume a in ].-infty,b .[; then a in { g where g is Element of REAL : g < b } by PROB_1:def 15; then consider h being Element of REAL such that A2: h = a & b > h; thus b > a by A2; end; assume a < b; then a in { g where g is Element of REAL : b > g } by A1; hence thesis by PROB_1:def 15; end; theorem Th18: for a being real number holds REAL \ {a} = ].-infty, a .[ \/ ]. a,+infty.[ proof let a be real number; A1: a is Real by XREAL_0:def 1; [. a,a .] = {a} by RCOMP_1:14; hence thesis by A1,LIMFUNC1:25; end; theorem Th19: for a, b, c, d being real number st a < b & b <= c holds [. a, b .] misses ]. c, d .] proof let a, b, c, d be real number; assume A1: a < b & b <= c; assume [. a, b .] meets ]. c, d .]; then consider x being set such that A2: x in [. a, b .] & x in ]. c, d .] by XBOOLE_0:3; reconsider x as real number by A2,XREAL_0:def 1; x <= b & x > c by A2,RCOMP_2:4,TOPREAL5:1; hence thesis by A1,AXIOMS:22; end; theorem Th20: for a, b, c, d being real number st a < b & b <= c holds [. a, b .[ misses [. c, d .] proof let a, b, c, d be real number; assume A1: a < b & b <= c; assume [. a, b .[ meets [. c, d .]; then consider x being set such that A2: x in [. a, b .[ & x in [. c, d .] by XBOOLE_0:3; reconsider x as real number by A2,XREAL_0:def 1; x < b & x >= c by A2,RCOMP_2:3,TOPREAL5:1; hence thesis by A1,AXIOMS:22; end; theorem for A, B being Subset of R^1, a, b, c, d being real number st a < b & b <= c & c < d & A = [. a, b .[ & B = ]. c, d .] holds A, B are_separated proof let A, B be Subset of R^1, a, b, c, d be real number; assume that A1: a < b & b <= c & c < d and A2: A = [. a, b .[ and A3: B = ]. c, d .]; Cl [. a, b .[ = [. a, b .] by A1,BORSUK_4:21; then Cl A = [. a, b .] by A2,Lm2; then A4: Cl A misses B by A1,A3,Th19; Cl ]. c, d .] = [. c, d .] by A1,BORSUK_4:20; then Cl B = [. c, d .] by A3,Lm2; then Cl B misses A by A1,A2,Th20; hence thesis by A4,CONNSP_1:def 1; end; theorem Th22: for a being real number holds REAL \ ].-infty, a .[ = [. a,+infty.[ proof let a be real number; a is Real by XREAL_0:def 1; hence thesis by LIMFUNC1:24; end; theorem Th23: for a being real number holds REAL \ ].-infty, a .] = ]. a,+infty.[ proof let a be real number; a is Real by XREAL_0:def 1; hence thesis by LIMFUNC1:24; end; theorem for a being real number holds REAL \ ]. a,+infty.[ = ].-infty, a .] proof let a be real number; a is Real by XREAL_0:def 1; hence thesis by LIMFUNC1:24; end; theorem Th25: for a being real number holds REAL \ [. a,+infty.[ = ].-infty, a .[ proof let a be real number; a is Real by XREAL_0:def 1; hence thesis by LIMFUNC1:24; end; theorem Th26: for a being real number holds ].-infty, a .] misses ]. a,+infty.[ proof let a be real number; REAL \ ].-infty, a .] = ]. a,+infty.[ by Th23; hence thesis by XBOOLE_1:79; end; theorem Th27: for a being real number holds ].-infty, a .[ misses [. a,+infty.[ proof let a be real number; REAL \ ].-infty, a .[ = [. a,+infty.[ by Th22; hence thesis by XBOOLE_1:79; end; theorem Th28: for a, b, c being real number st a <= c & c <= b holds [.a,b.] \/ [.c,+infty.[ = [.a,+infty.[ proof let a, b, c be real number; assume A1: a <= c & c <= b; A2: [.a,+infty.[ c= [.a,b.] \/ [.c,+infty.[ proof let r be set; assume A3: r in [.a,+infty.[; then reconsider s = r as Real; A4: a <= s by A3,Th15; per cases; suppose s <= b; then s in [.a,b.] by A4,TOPREAL5:1; hence thesis by XBOOLE_0:def 2; suppose not s <= b; then c <= s by A1,AXIOMS:22; then s in [.c,+infty.[ by Th15; hence thesis by XBOOLE_0:def 2; end; A5: a is Real by XREAL_0:def 1; A6: b is Real by XREAL_0:def 1; A7: c is Real by XREAL_0:def 1; [.a,b.] \/ [.c,+infty.[ c= [.a,+infty.[ proof A8: [.a,b.] c= [.a,+infty.[ by A5,A6,LIMFUNC1:12; [.c,+infty.[ c= [.a,+infty.[ by A1,A5,A7,LIMFUNC1:9; hence thesis by A8,XBOOLE_1:8; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th29: for a, b, c being real number st a <= c & c <= b holds ].-infty, c .] \/ [. a, b .] = ].-infty, b .] proof let a, b, c be real number; assume A1: a <= c & c <= b; thus ].-infty, c .] \/ [. a, b .] c= ].-infty, b .] proof let x be set; assume A2: x in ].-infty, c .] \/ [. a, b .]; then reconsider x as real number by XREAL_0:def 1; per cases by A2,XBOOLE_0:def 2; suppose x in ].-infty, c .]; then x <= c by Th16; then x <= b by A1,AXIOMS:22; hence thesis by Th16; suppose x in [. a, b .]; then x <= b by TOPREAL5:1; hence thesis by Th16; end; let x be set; assume A3: x in ].-infty, b .]; then reconsider x as real number by XREAL_0:def 1; per cases; suppose x <= c; then x in ].-infty, c .] by Th16; hence thesis by XBOOLE_0:def 2; suppose x > c; then A4: x > a by A1,AXIOMS:22; x <= b by A3,Th16; then x in [. a, b .] by A4,TOPREAL5:1; hence thesis by XBOOLE_0:def 2; end; theorem Th30: for T being 1-sorted, A being Subset of T holds { A } is Subset-Family of T by MEASURE1:6; theorem for T being 1-sorted, A, B being Subset of T holds { A, B } is Subset-Family of T by MEASURE1:7; theorem for T being 1-sorted, A, B, C being Subset of T holds { A, B, C } is Subset-Family of T by MEASURE1:8; definition cluster -> real Element of RAT; coherence; end; definition cluster -> real Element of RealSpace; coherence by METRIC_1:def 14,XREAL_0:def 1; end; theorem Th33: for A being Subset of R^1, p being Point of RealSpace holds p in Cl A iff for r being real number st r > 0 holds Ball (p, r) meets A by HAUSDORF:7,TOPMETR:def 7; theorem Th34: for p, q being Element of RealSpace st q >= p holds dist (p, q) = q - p proof let p, q be Element of RealSpace; assume p <= q; then A1: q - p >= 0 by SQUARE_1:12; dist (p, q) = abs (q - p) by TOPMETR:15 .= q - p by A1,ABSVALUE:def 1; hence thesis; end; Lm3: for r being real number holds r is Element of RealSpace by METRIC_1:def 14 ,XREAL_0:def 1; theorem Th35: for A being Subset of R^1 st A = RAT holds Cl A = the carrier of R^1 proof let A be Subset of R^1; assume A1: A = RAT; Cl A = the carrier of R^1 proof thus Cl A c= the carrier of R^1; thus the carrier of R^1 c= Cl A proof let x be set; assume x in the carrier of R^1; then reconsider p = x as Element of RealSpace by METRIC_1:def 14,TOPMETR:24; now let r be real number; assume A2: r > 0; reconsider pr = p + r as Element of RealSpace by Lm3; p < pr by A2,REAL_1:69; then consider Q being rational number such that A3: p < Q & Q < pr by RAT_1:22; A4: Q in A by A1,RAT_1:def 2; reconsider P = Q as Element of RealSpace by Lm3; P - p < pr - p by A3,REAL_1:54; then P - p < r by XCMPLX_1:26; then dist (p, P) < r by A3,Th34; then P in Ball (p, r) by METRIC_1:12; hence Ball (p, r) meets A by A4,XBOOLE_0:3; end; hence thesis by Th33; end; end; hence thesis; end; theorem Th36: for A being Subset of R^1, a, b being real number st A = ]. a, b .[ & a <> b holds Cl A = [. a, b .] proof let A be Subset of R^1, a, b be real number; assume A1: A = ]. a, b .[ & a <> b; then Cl ]. a, b .[ = [. a, b .] by TOPREAL6:82; hence thesis by A1,Lm2; end; begin :: Rational and Irrational Numbers definition cluster number_e -> irrational; coherence by IRRAT_1:42; end; definition func IRRAT -> Subset of REAL equals :Def3: REAL \ RAT; coherence by XBOOLE_1:36; end; definition let a, b be real number; func RAT (a, b) -> Subset of REAL equals :Def4: RAT /\ ]. a, b .[; coherence proof RAT /\ ]. a, b .[ c= ]. a, b .[ by XBOOLE_1:17; hence thesis by XBOOLE_1:1; end; func IRRAT (a, b) -> Subset of REAL equals :Def5: IRRAT /\ ]. a, b .[; coherence; end; theorem Th37: for x being real number holds x is irrational iff x in IRRAT proof let x be real number; A1: x in REAL by XREAL_0:def 1; hereby assume x is irrational; then not x in RAT by RAT_1:def 2; hence x in IRRAT by A1,Def3,XBOOLE_0:def 4; end; assume x in IRRAT; then not x in RAT by Def3,XBOOLE_0:def 4; hence thesis by RAT_1:def 2; end; definition cluster irrational (real number); existence by IRRAT_1:42; end; definition cluster IRRAT -> non empty; coherence by Th37,IRRAT_1:42; end; theorem Th38: for a being rational number, b being irrational (real number) holds a + b is irrational proof let a be rational number, b be irrational (real number); assume a + b is rational; then consider m, n being Integer such that A1: n <> 0 & a + b = m / n by RAT_1:3; consider m1, n1 being Integer such that A2: n1 <> 0 & a = m1 / n1 by RAT_1:3; A3: a + b - a = b by XCMPLX_1:26; a + b - a = (m*n1 - m1*n) / (n1*n) by A1,A2,XCMPLX_1:131; then b in RAT by A3; hence thesis by RAT_1:def 2; end; theorem Th39: for a being irrational (real number) holds -a is irrational proof let a be irrational (real number); assume -a is rational; then reconsider b = -a as rational number; -b is rational; hence thesis; end; theorem Th40: for a being rational number, b being irrational (real number) holds a - b is irrational proof let a be rational number, b be irrational (real number); -b is irrational by Th39; then a + -b is irrational by Th38; hence thesis by XCMPLX_0:def 8; end; theorem Th41: for a being rational number, b being irrational (real number) holds b - a is irrational proof let a be rational number, b be irrational (real number); b + -a is irrational by Th38; hence thesis by XCMPLX_0:def 8; end; theorem Th42: for a being rational number, b being irrational (real number) st a <> 0 holds a * b is irrational proof let a be rational number, b be irrational (real number); assume A1: a <> 0; assume a * b is rational; then consider m, n being Integer such that A2: n <> 0 & a * b = m / n by RAT_1:3; consider m1, n1 being Integer such that A3: n1 <> 0 & a = m1 / n1 by RAT_1:3; A4: a * b / a = b by A1,XCMPLX_1:90; a * b / a = (m * n1) / (n * m1) by A2,A3,XCMPLX_1:85; then b in RAT by A4; hence thesis by RAT_1:def 2; end; theorem Th43: for a being rational number, b being irrational (real number) st a <> 0 holds b / a is irrational proof let a be rational number, b be irrational (real number); assume A1: a <> 0; assume b / a is rational; then reconsider c = b / a as rational number; c * a is rational; hence thesis by A1,XCMPLX_1:88; end; definition cluster irrational -> non zero (real number); coherence proof let a be real number; assume A1: a is irrational; assume a is zero; then a = 0 / 1; hence thesis by A1; end; end; theorem Th44: for a being rational number, b being irrational (real number) st a <> 0 holds a / b is irrational proof let a be rational number, b be irrational (real number); assume A1: a <> 0; assume a / b is rational; then reconsider c = a / b as rational number; b <> 0 & c <> 0 by A1,XCMPLX_1:50; then c * b is irrational by Th42; hence thesis by XCMPLX_1:88; end; theorem Th45: for r being irrational (real number) holds frac r is irrational proof let r be irrational (real number); frac r = r - [\ r /] by INT_1:def 6; hence thesis by Th41; end; definition let r be irrational (real number); cluster frac r -> irrational; coherence by Th45; end; definition let a be irrational (real number); cluster -a -> irrational; coherence by Th39; end; definition let a be rational number, b be irrational (real number); cluster a + b -> irrational; coherence by Th38; cluster b + a -> irrational; coherence by Th38; cluster a - b -> irrational; coherence by Th40; cluster b - a -> irrational; coherence by Th41; end; definition cluster non zero (rational number); existence proof reconsider a = 1 as rational number; a <> 0; hence thesis; end; end; definition let a be non zero (rational number), b be irrational (real number); cluster a * b -> irrational; coherence by Th42; cluster b * a -> irrational; coherence by Th42; cluster a / b -> irrational; coherence by Th44; cluster b / a -> irrational; coherence by Th43; end; theorem for r being irrational (real number) holds 0 < frac r by INT_1:69; theorem Th47: for a, b being real number st a < b ex p1, p2 being rational number st a < p1 & p1 < p2 & p2 < b proof let a, b be real number; assume a < b; then consider p1 being rational number such that A1: a < p1 & p1 < b by RAT_1:22; consider p2 being rational number such that A2: p1 < p2 & p2 < b by A1,RAT_1:22; thus thesis by A1,A2; end; Lm4: for p1, p2, x being real number holds (1 - x) * p1 + x * p2 = p1 + x * (p2 - p1) proof let p1, p2, x be real number; thus (1 - x) * p1 + x * p2 = 1 * p1 - x * p1 + x * p2 by XCMPLX_1:40 .= 1 * p1 + - x * p1 + x * p2 by XCMPLX_0:def 8 .= 1 * p1 + (- x * p1 + x * p2) by XCMPLX_1:1 .= 1 * p1 + (x * p2 - x * p1) by XCMPLX_0:def 8 .= p1 + x * (p2 - p1) by XCMPLX_1:40; end; theorem Th48: for s1, s3, s4, l being real number holds s1 <= s3 & s1 < s4 & 0 < l & l < 1 implies s1 < (1-l)*s3+l*s4 proof let s1, s3, s4, l be real number; assume A1: s1 <= s3 & s1 < s4 & 0 < l & l < 1; then A2: l*s1 < l*s4 by REAL_1:70; 1-l > 0 by A1,SQUARE_1:11; then A3: (1-l)*s1 <= (1-l)*s3 by A1,AXIOMS:25; (1-l)*s1+l*s1 = (1-l+l)*s1 by XCMPLX_1:8 .= 1 * s1 by XCMPLX_1:27 .= s1; hence thesis by A2,A3,REAL_1:67; end; theorem Th49: for s1, s3, s4, l being real number holds s3 < s1 & s4 <= s1 & 0 < l & l < 1 implies (1-l)*s3+l*s4 < s1 proof let s1, s3, s4, l be real number; assume A1: s1 > s3 & s1 >= s4 & 0 < l & l < 1; then A2: l*s1 >= l*s4 by AXIOMS:25; 1-l > 0 by A1,SQUARE_1:11; then A3: (1-l)*s1 > (1-l)*s3 by A1,REAL_1:70; (1-l)*s1+l*s1 = (1-l+l)*s1 by XCMPLX_1:8 .= 1 * s1 by XCMPLX_1:27 .= s1; hence thesis by A2,A3,REAL_1:67; end; theorem Th50: for a, b being real number st a < b ex p being irrational (real number) st a < p & p < b proof let a, b be real number; assume a < b; then consider p1, p2 being rational number such that A1: a < p1 & p1 < p2 & p2 < b by Th47; A2: p2 - p1 <> 0 by A1,XCMPLX_1:15; set x = frac number_e; set y = (1 - x) * p1 + x * p2; A3: y = p1 + x * (p2 - p1) by Lm4; x * (p2 - p1) is irrational by A2,Th42; then A4: y is irrational by A3,Th38; A5: 0 < x & x < 1 by INT_1:69; then y < p2 by A1,Th49; then A6: y < b by A1,AXIOMS:22; y > p1 by A1,A5,Th48; then y > a by A1,AXIOMS:22; hence thesis by A4,A6; end; theorem Th51: for A being Subset of R^1 st A = IRRAT holds Cl A = the carrier of R^1 proof let A be Subset of R^1; assume A1: A = IRRAT; Cl A = the carrier of R^1 proof thus Cl A c= the carrier of R^1; thus the carrier of R^1 c= Cl A proof let x be set; assume x in the carrier of R^1; then reconsider p = x as Element of RealSpace by METRIC_1:def 14,TOPMETR:24; now let r be real number; assume A2: r > 0; reconsider pr = p + r as Element of RealSpace by Lm3; p < pr by A2,REAL_1:69; then consider Q being irrational (real number) such that A3: p < Q & Q < pr by Th50; A4: Q in A by A1,Th37; reconsider P = Q as Element of RealSpace by Lm3; P - p < pr - p by A3,REAL_1:54; then P - p < r by XCMPLX_1:26; then dist (p, P) < r by A3,Th34; then P in Ball (p, r) by METRIC_1:12; hence Ball (p, r) meets A by A4,XBOOLE_0:3; end; hence thesis by Th33; end; end; hence thesis; end; Lm5: for A being Subset of R^1, a, b being real number st a < b & A = RAT (a, b) holds a in Cl A & b in Cl A proof let A be Subset of R^1, a, b be real number; assume that A1: a < b and A2: A = RAT (a, b); thus a in Cl A proof reconsider a' = a as Element of RealSpace by Lm3; for r being real number st r > 0 holds Ball (a', r) meets A proof let r be real number; assume A3: r > 0; set p = a; reconsider pp = a + r as Element of RealSpace by Lm3; set pr = min (pp, (p + b)/2); A4: pr <= (p + b)/2 by SQUARE_1:35; A5: (p + b)/2 < b by A1,TOPREAL3:3; p < pr proof per cases by SQUARE_1:38; suppose pr = pp; hence thesis by A3,REAL_1:69; suppose pr = (p + b)/2; hence thesis by A1,TOPREAL3:3; end; then consider Q being rational number such that A6: p < Q & Q < pr by RAT_1:22; A7: Q in RAT by RAT_1:def 2; pr < b by A4,A5,AXIOMS:22; then a < Q & Q < b by A6,AXIOMS:22; then Q in ]. a, b .[ by JORDAN6:45; then Q in RAT /\ ]. a, b .[ by A7,XBOOLE_0:def 3; then A8: Q in A by A2,Def4; reconsider P = Q as Element of RealSpace by Lm3; A9: P - p < pr - p by A6,REAL_1:54; pr - p <= r proof per cases by SQUARE_1:38; suppose pr = pp; hence thesis by XCMPLX_1:26; suppose pr = (p + b)/2; pr <= pp by SQUARE_1:35; then pr - p <= pp - p by REAL_1:49; hence thesis by XCMPLX_1:26; end; then P - p < r by A9,AXIOMS:22; then dist (a', P) < r by A6,Th34; then P in Ball (a', r) by METRIC_1:12; hence thesis by A8,XBOOLE_0:3; end; hence thesis by Th33; end; b in Cl A proof reconsider a' = b as Element of RealSpace by Lm3; for r being real number st r > 0 holds Ball (a', r) meets A proof let r be real number; assume A10: r > 0; set p = b; reconsider pp = b - r as Element of RealSpace by Lm3; set pr = max (pp, (p + a)/2); A11: b < b + r by A10,REAL_1:69; A12: pr >= (p + a)/2 by SQUARE_1:46; A13: (p + a)/2 > a by A1,TOPREAL3:3; p > pr proof per cases by SQUARE_1:49; suppose pr = pp; hence thesis by A11,REAL_1:84; suppose pr = (p + a)/2; hence thesis by A1,TOPREAL3:3; end; then consider Q being rational number such that A14: pr < Q & Q < p by RAT_1:22; A15: Q in RAT by RAT_1:def 2; a < pr by A12,A13,AXIOMS:22; then a < Q & Q < b by A14,AXIOMS:22; then Q in ]. a, b .[ by JORDAN6:45; then Q in RAT /\ ]. a, b .[ by A15,XBOOLE_0:def 3; then A16: Q in A by A2,Def4; reconsider P = Q as Element of RealSpace by Lm3; A17: p - P < p - pr by A14,REAL_2:105; p - pr <= r proof per cases by SQUARE_1:49; suppose pr = pp; hence thesis by XCMPLX_1:18; suppose pr = (p + a)/2; pr >= pp by SQUARE_1:46; then p - pr <= p - pp by REAL_1:92; hence thesis by XCMPLX_1:18; end; then p - P < r by A17,AXIOMS:22; then dist (a', P) < r by A14,Th34; then P in Ball (a', r) by METRIC_1:12; hence thesis by A16,XBOOLE_0:3; end; hence thesis by Th33; end; hence thesis; end; Lm6: for A being Subset of R^1, a, b being real number st a < b & A = IRRAT (a, b) holds a in Cl A & b in Cl A proof let A be Subset of R^1, a, b be real number; assume that A1: a < b and A2: A = IRRAT (a, b); thus a in Cl A proof reconsider a' = a as Element of RealSpace by Lm3; for r being real number st r > 0 holds Ball (a', r) meets A proof let r be real number; assume A3: r > 0; set p = a; reconsider pp = a + r as Element of RealSpace by Lm3; set pr = min (pp, (p + b)/2); A4: pr <= (p + b)/2 by SQUARE_1:35; A5: (p + b)/2 < b by A1,TOPREAL3:3; p < pr proof per cases by SQUARE_1:38; suppose pr = pp; hence thesis by A3,REAL_1:69; suppose pr = (p + b)/2; hence thesis by A1,TOPREAL3:3; end; then consider Q being irrational (real number) such that A6: p < Q & Q < pr by Th50; A7: Q in IRRAT by Th37; pr < b by A4,A5,AXIOMS:22; then a < Q & Q < b by A6,AXIOMS:22; then Q in ]. a, b .[ by JORDAN6:45; then Q in IRRAT /\ ]. a, b .[ by A7,XBOOLE_0:def 3; then A8: Q in A by A2,Def5; reconsider P = Q as Element of RealSpace by Lm3; A9: P - p < pr - p by A6,REAL_1:54; pr - p <= r proof per cases by SQUARE_1:38; suppose pr = pp; hence thesis by XCMPLX_1:26; suppose pr = (p + b)/2; pr <= pp by SQUARE_1:35; then pr - p <= pp - p by REAL_1:49; hence thesis by XCMPLX_1:26; end; then P - p < r by A9,AXIOMS:22; then dist (a', P) < r by A6,Th34; then P in Ball (a', r) by METRIC_1:12; hence thesis by A8,XBOOLE_0:3; end; hence thesis by Th33; end; b in Cl A proof reconsider a' = b as Element of RealSpace by Lm3; for r being real number st r > 0 holds Ball (a', r) meets A proof let r be real number; assume A10: r > 0; set p = b; reconsider pp = b - r as Element of RealSpace by Lm3; set pr = max (pp, (p + a)/2); A11: b < b + r by A10,REAL_1:69; A12: pr >= (p + a)/2 by SQUARE_1:46; A13: (p + a)/2 > a by A1,TOPREAL3:3; p > pr proof per cases by SQUARE_1:49; suppose pr = pp; hence thesis by A11,REAL_1:84; suppose pr = (p + a)/2; hence thesis by A1,TOPREAL3:3; end; then consider Q being irrational (real number) such that A14: pr < Q & Q < p by Th50; A15: Q in IRRAT by Th37; a < pr by A12,A13,AXIOMS:22; then a < Q & Q < b by A14,AXIOMS:22; then Q in ]. a, b .[ by JORDAN6:45; then Q in IRRAT /\ ]. a, b .[ by A15,XBOOLE_0:def 3; then A16: Q in A by A2,Def5; reconsider P = Q as Element of RealSpace by Lm3; A17: p - P < p - pr by A14,REAL_2:105; p - pr <= r proof per cases by SQUARE_1:49; suppose pr = pp; hence thesis by XCMPLX_1:18; suppose pr = (p + a)/2; pr >= pp by SQUARE_1:46; then p - pr <= p - pp by REAL_1:92; hence thesis by XCMPLX_1:18; end; then p - P < r by A17,AXIOMS:22; then dist (a', P) < r by A14,Th34; then P in Ball (a', r) by METRIC_1:12; hence thesis by A16,XBOOLE_0:3; end; hence thesis by Th33; end; hence thesis; end; theorem Th52: for a, b, c being real number st a < b holds c in RAT (a,b) iff c is rational & a < c & c < b proof let a, b, c be real number; assume a < b; hereby assume c in RAT (a,b); then c in RAT /\ ]. a, b .[ by Def4; then c in RAT & c in ]. a, b .[ by XBOOLE_0:def 3; hence c is rational & a < c & c < b by JORDAN6:45,RAT_1:def 2; end; assume c is rational & a < c & c < b; then c in RAT & c in ]. a, b .[ by JORDAN6:45,RAT_1:def 2; then c in RAT /\ ]. a, b .[ by XBOOLE_0:def 3; hence thesis by Def4; end; theorem Th53: for a, b, c being real number st a < b holds c in IRRAT (a,b) iff c is irrational & a < c & c < b proof let a, b, c be real number; assume a < b; hereby assume c in IRRAT (a,b); then c in IRRAT /\ ]. a, b .[ by Def5; then c in IRRAT & c in ]. a, b .[ by XBOOLE_0:def 3; hence c is irrational & a < c & c < b by Th37,JORDAN6:45; end; assume c is irrational & a < c & c < b; then c in IRRAT & c in ]. a, b .[ by Th37,JORDAN6:45; then c in IRRAT /\ ]. a, b .[ by XBOOLE_0:def 3; hence thesis by Def5; end; theorem Th54: for A being Subset of R^1, a, b being real number st a < b & A = RAT (a, b) holds Cl A = [. a, b .] proof let A be Subset of R^1, a, b be real number; assume that A1: a < b and A2: A = RAT (a, b); RAT /\ ]. a, b .[ c= ]. a, b .[ by XBOOLE_1:17; then RAT /\ ]. a, b .[ c= the carrier of R^1 by TOPMETR:24,XBOOLE_1:1; then reconsider RR = RAT /\ ]. a, b .[ as Subset of R^1; reconsider ab = ]. a, b .[, RT = RAT as Subset of R^1 by RAT_1:4,TOPMETR:24; A3: Cl RR c= (Cl RT) /\ Cl ab by PRE_TOPC:51; A4: (the carrier of R^1) /\ Cl ab = Cl ab by XBOOLE_1:28; thus Cl A c= [. a, b .] proof let x be set; assume x in Cl A; then x in Cl RR by A2,Def4; then x in (Cl RT) /\ Cl ab by A3; then x in (the carrier of R^1) /\ Cl ab by Th35; hence thesis by A1,A4,Th36; end; thus [. a, b .] c= Cl A proof let x be set; assume A5: x in [. a, b .]; then x is real number by XREAL_0:def 1; then reconsider p = x as Element of RealSpace by Lm3; A6: a <= p by A5,TOPREAL5:1; A7: p <= b by A5,TOPREAL5:1; per cases by A7,REAL_1:def 5; suppose A8: p < b; now let r be real number; assume A9: r > 0; reconsider pp = p + r as Element of RealSpace by Lm3; set pr = min (pp, (p + b)/2); A10: pr <= (p + b)/2 by SQUARE_1:35; A11: (p + b)/2 < b by A8,TOPREAL3:3; p < pr proof per cases by SQUARE_1:38; suppose pr = pp; hence thesis by A9,REAL_1:69; suppose pr = (p + b)/2; hence thesis by A8,TOPREAL3:3; end; then consider Q being rational number such that A12: p < Q & Q < pr by RAT_1:22; A13: Q in RAT by RAT_1:def 2; pr < b by A10,A11,AXIOMS:22; then a < Q & Q < b by A6,A12,AXIOMS:22; then Q in ]. a, b .[ by JORDAN6:45; then Q in RAT /\ ]. a, b .[ by A13,XBOOLE_0:def 3; then A14: Q in A by A2,Def4; reconsider P = Q as Element of RealSpace by Lm3; A15: P - p < pr - p by A12,REAL_1:54; pr - p <= r proof per cases by SQUARE_1:38; suppose pr = pp; hence thesis by XCMPLX_1:26; suppose pr = (p + b)/2; pr <= pp by SQUARE_1:35; then pr - p <= pp - p by REAL_1:49; hence thesis by XCMPLX_1:26; end; then P - p < r by A15,AXIOMS:22; then dist (p, P) < r by A12,Th34; then P in Ball (p, r) by METRIC_1:12; hence Ball (p, r) meets A by A14,XBOOLE_0:3; end; hence thesis by Th33; suppose p = b; hence thesis by A1,A2,Lm5; end; end; theorem Th55: for A being Subset of R^1, a, b being real number st a < b & A = IRRAT (a, b) holds Cl A = [. a, b .] proof let A be Subset of R^1, a, b be real number; assume that A1: a < b and A2: A = IRRAT (a, b); reconsider RR = IRRAT /\ ]. a, b .[ as Subset of R^1 by TOPMETR:24; reconsider ab = ]. a, b .[, RT = IRRAT as Subset of R^1 by TOPMETR:24; A3: Cl RR c= (Cl RT) /\ Cl ab by PRE_TOPC:51; A4: (the carrier of R^1) /\ Cl ab = Cl ab by XBOOLE_1:28; thus Cl A c= [. a, b .] proof let x be set; assume x in Cl A; then x in Cl RR by A2,Def5; then x in (Cl RT) /\ Cl ab by A3; then x in (the carrier of R^1) /\ Cl ab by Th51; hence thesis by A1,A4,Th36; end; thus [. a, b .] c= Cl A proof let x be set; assume A5: x in [. a, b .]; then x is real number by XREAL_0:def 1; then reconsider p = x as Element of RealSpace by Lm3; A6: a <= p by A5,TOPREAL5:1; A7: p <= b by A5,TOPREAL5:1; per cases by A7,REAL_1:def 5; suppose A8: p < b; now let r be real number; assume A9: r > 0; reconsider pp = p + r as Element of RealSpace by Lm3; set pr = min (pp, (p + b)/2); A10: pr <= (p + b)/2 by SQUARE_1:35; A11: (p + b)/2 < b by A8,TOPREAL3:3; p < pr proof per cases by SQUARE_1:38; suppose pr = pp; hence thesis by A9,REAL_1:69; suppose pr = (p + b)/2; hence thesis by A8,TOPREAL3:3; end; then consider Q being irrational (real number) such that A12: p < Q & Q < pr by Th50; A13: Q in IRRAT by Th37; pr < b by A10,A11,AXIOMS:22; then a < Q & Q < b by A6,A12,AXIOMS:22; then Q in ]. a, b .[ by JORDAN6:45; then Q in IRRAT /\ ]. a, b .[ by A13,XBOOLE_0:def 3; then A14: Q in A by A2,Def5; reconsider P = Q as Element of RealSpace by Lm3; A15: P - p < pr - p by A12,REAL_1:54; pr - p <= r proof per cases by SQUARE_1:38; suppose pr = pp; hence thesis by XCMPLX_1:26; suppose pr = (p + b)/2; pr <= pp by SQUARE_1:35; then pr - p <= pp - p by REAL_1:49; hence thesis by XCMPLX_1:26; end; then P - p < r by A15,AXIOMS:22; then dist (p, P) < r by A12,Th34; then P in Ball (p, r) by METRIC_1:12; hence Ball (p, r) meets A by A14,XBOOLE_0:3; end; hence thesis by Th33; suppose p = b; hence thesis by A1,A2,Lm6; end; end; theorem Th56: for T being connected TopSpace, A being closed open Subset of T holds A = {} or A = [#]T proof let T be connected TopSpace, A be closed open Subset of T; assume that A1: A <> {} and A2: A <> [#]T; A3: A <> {}T by A1; A4: [#]T = A \/ A` by PRE_TOPC:18; A misses A` by TOPS_1:20; then A5: A, A` are_separated by A4,CONNSP_1:3; [#]T \ A <> {} by A2,PRE_TOPC:23; then A` <> {} by PRE_TOPC:17; then not [#]T is connected by A3,A4,A5,CONNSP_1:16; hence thesis by CONNSP_1:28; end; theorem Th57: for A being Subset of R^1 st A is closed open holds A = {} or A = REAL proof let A be Subset of R^1; assume A is closed open; then reconsider A as closed open Subset of R^1; A = {} or A = [#]R^1 by Th56; hence thesis by PRE_TOPC:def 3,TOPMETR:24; end; begin :: Topological Properties of Intervals theorem Th58: for A being Subset of R^1, a, b being real number st A = [. a, b .[ & a <> b holds Cl A = [. a, b .] proof let A be Subset of R^1, a, b be real number; assume A1: A = [. a, b .[ & a <> b; then Cl [. a, b .[ = [. a, b .] by BORSUK_4:21; hence thesis by A1,Lm2; end; theorem Th59: for A being Subset of R^1, a, b being real number st A = ]. a, b .] & a <> b holds Cl A = [. a, b .] proof let A be Subset of R^1, a, b be real number; assume A1: A = ]. a, b .] & a <> b; then Cl ]. a, b .] = [. a, b .] by BORSUK_4:20; hence thesis by A1,Lm2; end; theorem for A being Subset of R^1, a, b, c being real number st A = [. a, b .[ \/ ]. b, c .] & a < b & b < c holds Cl A = [. a, c .] proof let A be Subset of R^1, a, b, c be real number; assume that A1: A = [. a, b .[ \/ ]. b, c .] and A2: a < b & b < c; reconsider B = [. a, b .[, C = ]. b, c .] as Subset of R^1 by TOPMETR:24; Cl A = Cl B \/ Cl C by A1,PRE_TOPC:50 .= [. a, b .] \/ Cl C by A2,Th58 .= [. a, b .] \/ [. b, c .] by A2,Th59 .= [. a, c .] by A2,TREAL_1:2; hence thesis; end; theorem Th61: for A being Subset of R^1, a being real number st A = {a} holds Cl A = {a} proof let A be Subset of R^1, a be real number; assume A1: A = {a}; a is Point of R^1 by TOPMETR:24,XREAL_0:def 1; hence thesis by A1,YELLOW_8:35; end; theorem Th62: for A being Subset of REAL, B being Subset of R^1 st A = B holds A is open iff B is open proof let A be Subset of REAL, B be Subset of R^1; assume A1: A = B; hereby assume A is open; then A in Family_open_set RealSpace by JORDAN5A:6; then A in the topology of TopSpaceMetr RealSpace by TOPMETR:16; hence B is open by A1,PRE_TOPC:def 5,TOPMETR:def 7; end; assume B is open; then B in the topology of R^1 by PRE_TOPC:def 5; then A in Family_open_set RealSpace by A1,TOPMETR:16,def 7; hence thesis by JORDAN5A:6; end; Lm7: for a being real number holds ]. a,+infty.[ is open proof let a be real number; a is Real by XREAL_0:def 1; hence thesis by FCONT_3:7; end; Lm8: for a being real number holds ].-infty,a.] is closed proof let a be real number; a is Real by XREAL_0:def 1; hence thesis by FCONT_3:6; end; Lm9: for a being real number holds ].-infty,a.[ is open proof let a be real number; a is Real by XREAL_0:def 1; hence thesis by FCONT_3:8; end; Lm10: for a being real number holds [.a,+infty.[ is closed proof let a be real number; a is Real by XREAL_0:def 1; hence thesis by FCONT_3:5; end; theorem Th63: for A being Subset of R^1, a being real number st A = ]. a,+infty.[ holds A is open proof let A be Subset of R^1, a be real number; assume A1: A = ]. a,+infty.[; ]. a,+infty.[ is open by Lm7; hence thesis by A1,Th62; end; theorem Th64: for A being Subset of R^1, a being real number st A = ].-infty,a .[ holds A is open proof let A be Subset of R^1, a be real number; assume A1: A = ].-infty,a .[; ].-infty,a .[ is open by Lm9; hence thesis by A1,Th62; end; theorem Th65: for A being Subset of R^1, a being real number st A = ].-infty,a.] holds A is closed proof let A be Subset of R^1, a be real number; assume A1: A = ].-infty,a.]; ].-infty,a.] is closed by Lm8; hence thesis by A1,TOPREAL6:79; end; theorem Th66: for A being Subset of R^1, a being real number st A = [. a,+infty.[ holds A is closed proof let A be Subset of R^1, a be real number; assume A1: A = [. a,+infty.[; [. a,+infty.[ is closed by Lm10; hence thesis by A1,TOPREAL6:79; end; theorem Th67: for a being real number holds [. a,+infty.[ = {a} \/ ]. a,+infty.[ proof let a be real number; thus [. a,+infty.[ c= {a} \/ ]. a,+infty.[ proof let x be set; assume A1: x in [. a,+infty.[; then reconsider x as real number by XREAL_0:def 1; A2: x >= a by A1,Th15; per cases by A2,REAL_1:def 5; suppose x = a; then x in {a} by TARSKI:def 1; hence thesis by XBOOLE_0:def 2; suppose x > a; then x in ]. a,+infty.[ by Th14; hence thesis by XBOOLE_0:def 2; end; let x be set; assume A3: x in {a} \/ ]. a,+infty.[; then reconsider x as real number by XREAL_0:def 1; x in {a} or x in ]. a,+infty.[ by A3,XBOOLE_0:def 2; then x = a or x > a by Th14,TARSKI:def 1; hence thesis by Th15; end; theorem Th68: for a being real number holds ].-infty,a.] = {a} \/ ].-infty,a.[ proof let a be real number; thus ].-infty,a.] c= {a} \/ ].-infty,a.[ proof let x be set; assume A1: x in ].-infty,a.]; then reconsider x as real number by XREAL_0:def 1; A2: x <= a by A1,Th16; per cases by A2,REAL_1:def 5; suppose x = a; then x in {a} by TARSKI:def 1; hence thesis by XBOOLE_0:def 2; suppose x < a; then x in ].-infty,a.[ by Th17; hence thesis by XBOOLE_0:def 2; end; let x be set; assume A3: x in {a} \/ ].-infty,a.[; then reconsider x as real number by XREAL_0:def 1; x in {a} or x in ].-infty,a.[ by A3,XBOOLE_0:def 2; then x = a or x < a by Th17,TARSKI:def 1; hence thesis by Th16; end; theorem Th69: for a being real number holds ]. a,+infty.[ c= [. a,+infty.[ proof let a be real number; a is Real by XREAL_0:def 1; hence thesis by LIMFUNC1:10; end; theorem Th70: for a being real number holds ].-infty,a .[ c= ].-infty,a .] proof let a be real number; a is Real by XREAL_0:def 1; hence thesis by LIMFUNC1:15; end; definition let a be real number; cluster ]. a,+infty.[ -> non empty; coherence proof a < a + 1 by REAL_1:69; hence thesis by Th14; end; cluster ].-infty,a .] -> non empty; coherence by Th16; cluster ].-infty,a .[ -> non empty; coherence proof a - 1 < a by INT_1:26; hence thesis by Th17; end; cluster [. a,+infty.[ -> non empty; coherence by Th15; end; theorem Th71: for a being real number holds ]. a,+infty.[ <> REAL proof let a be real number; a in REAL by XREAL_0:def 1; hence thesis by Th14; end; theorem for a being real number holds [. a,+infty.[ <> REAL proof let a be real number; A1: a - 1 in REAL by XREAL_0:def 1; a - 1 < a by INT_1:26; hence thesis by A1,Th15; end; theorem for a being real number holds ].-infty, a .] <> REAL proof let a be real number; A1: a + 1 in REAL by XREAL_0:def 1; a + 1 > a by REAL_1:69; hence thesis by A1,Th16; end; theorem Th74: for a being real number holds ].-infty, a .[ <> REAL proof let a be real number; A1: a + 1 in REAL by XREAL_0:def 1; a + 1 > a by REAL_1:69; hence thesis by A1,Th17; end; theorem Th75: for A being Subset of R^1, a being real number st A = ]. a,+infty.[ holds Cl A = [. a,+infty.[ proof let A be Subset of R^1, a be real number; reconsider A' = A as Subset of R^1; assume A1: A = ]. a,+infty.[; then A2: A' is open by Th63; reconsider C = [. a,+infty.[ as Subset of R^1 by TOPMETR:24; A3: C is closed by Th66; A4: C = A' \/ {a} by A1,Th67; A5: A' c= Cl A' by PRE_TOPC:48; A' c= C by A1,Th69; then A6: Cl A' c= C by A3,TOPS_1:31; per cases by A4,A5,A6,Th2; suppose Cl A' = C; hence thesis; suppose Cl A' = A'; then A' = {} or A' = REAL by A2,Th57; hence thesis by A1,Th71; end; theorem for a being real number holds Cl ]. a,+infty.[ = [. a,+infty.[ proof let a be real number; reconsider A = ]. a,+infty.[ as Subset of R^1 by TOPMETR:24; Cl A = [. a,+infty.[ by Th75; hence thesis by TOPREAL6:80; end; theorem Th77: for A being Subset of R^1, a being real number st A = ].-infty, a .[ holds Cl A = ].-infty, a .] proof let A be Subset of R^1, a be real number; reconsider A' = A as Subset of R^1; assume A1: A = ].-infty, a .[; then A2: A' is open by Th64; reconsider C = ].-infty, a .] as Subset of R^1 by TOPMETR:24; A3: C is closed by Th65; A4: C = A' \/ {a} by A1,Th68; A5: A' c= Cl A' by PRE_TOPC:48; A' c= C by A1,Th70; then A6: Cl A' c= C by A3,TOPS_1:31; per cases by A4,A5,A6,Th2; suppose Cl A' = C; hence thesis; suppose Cl A' = A'; then A' = {} or A' = REAL by A2,Th57; hence thesis by A1,Th74; end; theorem for a being real number holds Cl ].-infty, a .[ = ].-infty, a .] proof let a be real number; reconsider A = ].-infty, a .[ as Subset of R^1 by TOPMETR:24; Cl A = ].-infty, a .] by Th77; hence thesis by TOPREAL6:80; end; theorem Th79: for A, B being Subset of R^1, b being real number st A = ].-infty, b .[ & B = ]. b,+infty.[ holds A, B are_separated proof let A, B be Subset of R^1, b be real number; assume that A1: A = ].-infty, b .[ and A2: B = ]. b,+infty.[; Cl A = ].-infty, b .] by A1,Th77; then A3: Cl A misses B by A2,Th26; Cl B = [. b,+infty.[ by A2,Th75; then Cl B misses A by A1,Th27; hence thesis by A3,CONNSP_1:def 1; end; theorem for A being Subset of R^1, a, b being real number st a < b & A = [. a, b .[ \/ ]. b,+infty.[ holds Cl A = [. a,+infty.[ proof let A be Subset of R^1, a, b be real number; assume A1: a < b & A = [. a, b .[ \/ ]. b,+infty.[; reconsider B = [. a, b .[, C = ]. b,+infty.[ as Subset of R^1 by TOPMETR:24; A2: Cl B = [. a, b .] by A1,Th58; A3: Cl C = [. b,+infty.[ by Th75; Cl A = Cl B \/ Cl C by A1,PRE_TOPC:50; hence thesis by A1,A2,A3,Th28; end; theorem Th81: for A being Subset of R^1, a, b being real number st a < b & A = ]. a, b .[ \/ ]. b,+infty.[ holds Cl A = [. a,+infty.[ proof let A be Subset of R^1, a, b be real number; assume A1: a < b & A = ]. a, b .[ \/ ]. b,+infty.[; reconsider B = ]. a, b .[, C = ]. b,+infty.[ as Subset of R^1 by TOPMETR:24; A2: Cl A = Cl B \/ Cl C by A1,PRE_TOPC:50; A3: Cl B = [. a, b .] by A1,Th36; Cl C = [. b,+infty.[ by Th75; hence thesis by A1,A2,A3,Th28; end; theorem for A being Subset of R^1, a, b, c being real number st a < b & b < c & A = RAT (a,b) \/ ]. b, c .[ \/ ]. c,+infty.[ holds Cl A = [. a,+infty.[ proof let A be Subset of R^1; let a, b, c be real number; assume A1: a < b & b < c; reconsider C = ]. b, c .[ \/ ]. c,+infty.[ as Subset of R^1 by TOPMETR:24; reconsider B = RAT (a,b) as Subset of R^1 by TOPMETR:24; assume A = RAT (a,b) \/ ]. b, c .[ \/ ]. c,+infty.[; then A = RAT (a,b) \/ C by XBOOLE_1:4; then Cl A = Cl B \/ Cl C by PRE_TOPC:50; then Cl A = Cl B \/ [. b,+infty.[ by A1,Th81; then Cl A = [. a, b .] \/ [. b,+infty.[ by A1,Th54; hence Cl A = [. a,+infty.[ by A1,Th28; end; theorem for A being Subset of R^1 holds A` = REAL \ A by Th13,PRE_TOPC:17; theorem Th84: for a, b being real number st a < b holds IRRAT (a, b) misses RAT (a, b) proof let a, b be real number; assume A1: a < b; assume IRRAT (a, b) meets RAT (a, b); then consider x being set such that A2: x in IRRAT (a, b) & x in RAT (a, b) by XBOOLE_0:3; reconsider x as real number by A2,XREAL_0:def 1; x is rational & x is irrational by A1,A2,Th52,Th53; hence thesis; end; theorem Th85: for a, b being real number st a < b holds REAL \ RAT (a, b) = ].-infty,a.] \/ IRRAT (a, b) \/ [.b,+infty.[ proof let a, b be real number; assume A1: a < b; thus REAL \ RAT (a, b) c= ].-infty,a.] \/ IRRAT (a, b) \/ [.b,+infty.[ proof let x be set; assume x in REAL \ RAT (a, b); then A2: x in REAL & not x in RAT (a, b) by XBOOLE_0:def 4; then reconsider x as real number by XREAL_0:def 1; per cases; suppose x <= a & x < b; then x in ].-infty,a.] by Th16; then x in ].-infty,a.] \/ IRRAT (a, b) by XBOOLE_0:def 2; hence thesis by XBOOLE_0:def 2; suppose x <= a & x >= b; then x in ].-infty,a.] by Th16; then x in ].-infty,a.] \/ IRRAT (a, b) by XBOOLE_0:def 2; hence thesis by XBOOLE_0:def 2; suppose A3: x > a & x < b; x in IRRAT (a, b) proof per cases; suppose x is rational; hence thesis by A1,A2,A3,Th52; suppose x is irrational; hence thesis by A1,A3,Th53; end; then x in ].-infty,a.] \/ IRRAT (a, b) by XBOOLE_0:def 2; hence thesis by XBOOLE_0:def 2; suppose x > a & x >= b; then x in [.b,+infty.[ by Th15; hence thesis by XBOOLE_0:def 2; end; let x be set; assume A4: x in ].-infty,a.] \/ IRRAT (a, b) \/ [.b,+infty.[; then reconsider x as real number by XREAL_0:def 1; A5: x in ].-infty,a.] \/ IRRAT (a, b) or x in [.b,+infty.[ by A4,XBOOLE_0:def 2; per cases by A5,XBOOLE_0:def 2; suppose x in ].-infty,a.]; then x <= a by Th16; then x in REAL & not x in RAT (a, b) by A1,Th52,XREAL_0:def 1; hence thesis by XBOOLE_0:def 4; suppose A6: x in IRRAT (a, b); A7: x in REAL by XREAL_0:def 1; IRRAT (a, b) misses RAT (a, b) by A1,Th84; then not x in RAT (a,b) by A6,XBOOLE_0:3; hence thesis by A7,XBOOLE_0:def 4; suppose x in [.b,+infty.[; then x >= b by Th15; then x in REAL & not x in RAT (a, b) by A1,Th52,XREAL_0:def 1; hence thesis by XBOOLE_0:def 4; end; theorem Th86: for a, b, c being real number st a <= b & b < c holds not a in ]. b, c .[ \/ ]. c,+infty.[ proof let a, b, c be real number; assume A1: a <= b & b < c; then a < c by AXIOMS:22; then not a in ]. b, c .[ & not a in ]. c,+infty.[ by A1,Th14,JORDAN6:45; hence thesis by XBOOLE_0:def 2; end; theorem Th87: for a, b being real number st a < b holds not b in ]. a, b .[ \/ ]. b,+infty.[ proof let a, b be real number; assume a < b; not b in ]. a, b .[ & not b in ]. b,+infty.[ by Th14,JORDAN6:45; hence thesis by XBOOLE_0:def 2; end; theorem Th88: for a, b being real number st a < b holds [.a,+infty.[ \ (]. a, b .[ \/ ]. b,+infty.[) = {a} \/ {b} proof let a, b be real number; assume A1: a < b; then A2: not a in ]. a,b .[ \/ ]. b,+infty.[ by Th86; not b in ]. a,b .[ \/ ]. b,+infty.[ by A1,Th87; then A3: {a, b} misses ]. a,b .[ \/ ]. b,+infty.[ by A2,ZFMISC_1:57; [. a,+infty.[ = [. a,b .] \/ [. b,+infty.[ by A1,Th28 .= {a, b} \/ ]. a,b .[ \/ [. b,+infty.[ by A1,RCOMP_1:11 .= {a, b} \/ ]. a,b .[ \/ ({b} \/ ]. b,+infty.[) by Th67 .= {a, b} \/ ]. a,b .[ \/ {b} \/ ]. b,+infty.[ by XBOOLE_1:4 .= {a, b} \/ {b} \/ ]. a,b .[ \/ ]. b,+infty.[ by XBOOLE_1:4 .= {a, b} \/ ]. a,b .[ \/ ]. b,+infty.[ by ZFMISC_1:14 .= {a, b} \/ (]. a,b .[ \/ ]. b,+infty.[) by XBOOLE_1:4; then [.a,+infty.[ \ (]. a, b .[ \/ ]. b,+infty.[) = {a, b} by A3,XBOOLE_1:88; hence thesis by ENUMSET1:41; end; theorem for A being Subset of R^1 st A = RAT (2,3) \/ ]. 3, 4 .[ \/ ]. 4,+infty.[ holds A` = ].-infty,2 .] \/ IRRAT(2,3) \/ {3} \/ {4} proof let A be Subset of R^1; assume A = RAT (2,3) \/ ]. 3, 4 .[ \/ ]. 4,+infty.[; then A1: A` = REAL \ (RAT (2,3) \/ ]. 3, 4 .[ \/ ]. 4,+infty.[) by Th13,PRE_TOPC:17 .= REAL \ (RAT (2,3) \/ (]. 3, 4 .[ \/ ]. 4,+infty.[)) by XBOOLE_1:4 .= REAL \ RAT (2,3) \ (]. 3, 4 .[ \/ ]. 4,+infty.[) by XBOOLE_1:41 .= (].-infty,2.] \/ IRRAT (2, 3) \/ [.3,+infty.[) \ (]. 3, 4 .[ \/ ]. 4,+infty.[) by Th85; A2: ].-infty,3.] misses ]. 3,+infty.[ by Th26; A3: ].-infty,2.] \/ IRRAT (2, 3) c= ].-infty,3.] proof let x be set; assume A4: x in ].-infty,2.] \/ IRRAT (2, 3); then reconsider x as real number by XREAL_0:def 1; per cases by A4,XBOOLE_0:def 2; suppose x in ].-infty,2.]; then x <= 2 by Th16; then x <= 3 by AXIOMS:22; hence thesis by Th16; suppose x in IRRAT (2, 3); then x < 3 by Th53; hence thesis by Th16; end; ]. 3, 4 .[ \/ ]. 4,+infty.[ c= ]. 3,+infty.[ proof let x be set; assume A5: x in ]. 3, 4 .[ \/ ]. 4,+infty.[; then reconsider x as real number by XREAL_0:def 1; per cases by A5,XBOOLE_0:def 2; suppose x in ]. 3, 4 .[; then x > 3 by JORDAN6:45; hence thesis by Th14; suppose x in ]. 4,+infty.[; then x > 4 by Th14; then x > 3 by AXIOMS:22; hence thesis by Th14; end; then ].-infty,2.] \/ IRRAT (2, 3) misses ]. 3, 4 .[ \/ ]. 4,+infty.[ by A2,A3,XBOOLE_1:64; then A` = ([.3,+infty.[ \ (]. 3, 4 .[ \/ ]. 4,+infty.[)) \/ (].-infty,2.] \/ IRRAT (2, 3)) by A1,XBOOLE_1:87 .= (].-infty,2.] \/ IRRAT (2, 3)) \/ ({3} \/ {4}) by Th88 .= ].-infty,2 .] \/ IRRAT(2,3) \/ {3} \/ {4} by XBOOLE_1:4; hence thesis; end; theorem for A being Subset of R^1, a being real number st A = {a} holds A` = ].-infty, a .[ \/ ]. a,+infty.[ proof let A be Subset of R^1, a be real number; assume A = {a}; then A` = REAL \ {a} by Th13,PRE_TOPC:17; hence thesis by Th18; end; Lm11: IRRAT(2,3) \/ {3} \/ {4} c= ]. 1,+infty.[ proof let x be set; assume A1: x in IRRAT(2,3) \/ {3} \/ {4}; then reconsider x as real number by XREAL_0:def 1; A2: x in IRRAT(2,3) \/ {3} or x in {4} by A1,XBOOLE_0:def 2; per cases by A2,XBOOLE_0:def 2; suppose x in IRRAT(2,3); then x > 2 by Th53; then x > 1 by AXIOMS:22; hence thesis by Th14; suppose x in {3}; then x = 3 by TARSKI:def 1; hence thesis by Th14; suppose x in {4}; then x = 4 by TARSKI:def 1; hence thesis by Th14; end; ]. 1,+infty.[ c= [. 1,+infty.[ by Th69; then Lm12: IRRAT(2,3) \/ {3} \/ {4} c= [. 1,+infty.[ by Lm11,XBOOLE_1:1; Lm13: ].-infty, 1 .[ /\ (].-infty, 2 .] \/ IRRAT(2,3) \/ {3} \/ {4}) = ].-infty, 1 .[ proof [. 1,+infty.[ misses ].-infty, 1 .[ by Th27; then A1: IRRAT(2,3) \/ {3} \/ {4} misses ].-infty, 1 .[ by Lm12,XBOOLE_1:63; A2: ].-infty, 1 .[ c= ].-infty, 2 .] proof let x be set; assume A3: x in ].-infty, 1 .[; then reconsider x as real number by XREAL_0:def 1; x < 1 by A3,Th17; then x < 2 by AXIOMS:22; hence thesis by Th16; end; ].-infty, 1 .[ /\ (].-infty, 2 .] \/ IRRAT(2,3) \/ {3} \/ {4}) = ].-infty, 1 .[ /\ (].-infty, 2 .] \/ (IRRAT(2,3) \/ {3} \/ {4})) by XBOOLE_1:113 .= ].-infty, 1 .[ /\ ].-infty, 2 .] by A1,XBOOLE_1:78 .= ].-infty, 1 .[ by A2,XBOOLE_1:28; hence thesis; end; theorem Th91: for a, b being real number st a < b holds ]. a,+infty.[ /\ ].-infty, b .] = ]. a, b .] proof let a, b be real number; assume a < b; thus ]. a,+infty.[ /\ ].-infty, b .] c= ]. a, b .] proof let x be set; assume A1: x in ]. a,+infty.[ /\ ].-infty, b .]; then reconsider x as real number by XREAL_0:def 1; x in ]. a,+infty.[ & x in ].-infty, b .] by A1,XBOOLE_0:def 3; then x > a & x <= b by Th14,Th16; hence thesis by RCOMP_2:4; end; let x be set; assume A2: x in ]. a, b .]; then reconsider x as real number by XREAL_0:def 1; a < x & x <= b by A2,RCOMP_2:4; then x in ]. a,+infty.[ & x in ].-infty, b .] by Th14,Th16; hence thesis by XBOOLE_0:def 3; end; Lm14: ]. 1,+infty.[ /\ (].-infty, 2 .] \/ IRRAT(2,3) \/ {3} \/ {4}) = ]. 1, 2 .] \/ IRRAT(2,3) \/ {3} \/ {4} proof ]. 1,+infty.[ /\ (].-infty, 2 .] \/ IRRAT(2,3) \/ {3} \/ {4}) = ]. 1,+infty.[ /\ (].-infty, 2 .] \/ (IRRAT(2,3) \/ {3} \/ {4})) by XBOOLE_1:113 .= (]. 1,+infty.[ /\ ].-infty, 2 .]) \/ (]. 1,+infty.[ /\ (IRRAT(2,3) \/ {3} \/ {4})) by XBOOLE_1:23 .= (]. 1,+infty.[ /\ ].-infty, 2 .]) \/ (IRRAT(2,3) \/ {3} \/ {4}) by Lm11,XBOOLE_1:28 .= ]. 1,2 .] \/ (IRRAT(2,3) \/ {3} \/ {4}) by Th91 .= ]. 1,2 .] \/ IRRAT(2,3) \/ {3} \/ {4} by XBOOLE_1:113; hence thesis; end; theorem (].-infty, 1 .[ \/ ]. 1,+infty.[) /\ (].-infty, 2 .] \/ IRRAT(2,3) \/ {3} \/ {4}) = ].-infty, 1 .[ \/ ]. 1, 2 .] \/ IRRAT(2,3) \/ {3} \/ {4} proof (].-infty, 1 .[ \/ ]. 1,+infty.[) /\ (].-infty, 2 .] \/ IRRAT(2,3) \/ {3} \/ {4}) = ].-infty, 1 .[ \/ (]. 1,+infty.[ /\ (].-infty, 2 .] \/ IRRAT(2,3) \/ {3} \/ {4})) by Lm13,XBOOLE_1:23 .= ].-infty, 1 .[ \/ (]. 1, 2 .] \/ IRRAT(2,3) \/ ({3} \/ {4})) by Lm14,XBOOLE_1:4 .= ].-infty, 1 .[ \/ ]. 1, 2 .] \/ IRRAT(2,3) \/ ({3} \/ {4}) by XBOOLE_1:113; hence thesis by XBOOLE_1:4; end; theorem Th93: for a, b being real number st a <= b holds ].-infty,b.[ \ {a} = ].-infty, a .[ \/ ]. a, b .[ proof let a, b be real number; assume A1: a <= b; thus ].-infty,b.[ \ {a} c= ].-infty, a .[ \/ ]. a, b .[ proof let x be set; assume A2: x in ].-infty,b .[ \ {a}; then reconsider x as real number by XREAL_0:def 1; A3: x in ].-infty,b.[ & x <> a by A2,ZFMISC_1:64; then A4: x < b by Th17; per cases by A3,REAL_1:def 5; suppose x < a; then x in ].-infty, a .[ by Th17; hence thesis by XBOOLE_0:def 2; suppose x > a; then x in ]. a, b .[ by A4,JORDAN6:45; hence thesis by XBOOLE_0:def 2; end; let x be set; assume A5: x in ].-infty, a .[ \/ ]. a, b .[; then reconsider x as real number by XREAL_0:def 1; per cases by A5,XBOOLE_0:def 2; suppose A6: x in ].-infty, a .[; then x < a by Th17; then x < b by A1,AXIOMS:22; then x in ].-infty,b.[ & x <> a by A6,Th17; hence thesis by ZFMISC_1:64; suppose x in ]. a, b .[; then a < x & x < b by JORDAN6:45; then x in ].-infty,b.[ & x <> a by Th17; hence thesis by ZFMISC_1:64; end; theorem Th94: for a, b being real number st a <= b holds ]. a,+infty.[ \ {b} = ]. a, b .[ \/ ]. b,+infty.[ proof let a, b be real number; assume A1: a <= b; thus ]. a,+infty.[ \ {b} c= ]. a, b .[ \/ ]. b,+infty.[ proof let x be set; assume A2: x in ]. a,+infty.[ \ {b}; then reconsider x as real number by XREAL_0:def 1; A3: x in ]. a,+infty.[ & x <> b by A2,ZFMISC_1:64; then A4: x > a by Th14; per cases by A3,REAL_1:def 5; suppose x > b; then x in ]. b,+infty.[ by Th14; hence thesis by XBOOLE_0:def 2; suppose x < b; then x in ]. a, b .[ by A4,JORDAN6:45; hence thesis by XBOOLE_0:def 2; end; let x be set; assume A5: x in ]. a, b .[ \/ ]. b,+infty.[; then reconsider x as real number by XREAL_0:def 1; per cases by A5,XBOOLE_0:def 2; suppose A6: x in ]. b,+infty.[; then x > b by Th14; then x > a by A1,AXIOMS:22; then x in ]. a,+infty.[ & x <> b by A6,Th14; hence thesis by ZFMISC_1:64; suppose x in ]. a, b .[; then a < x & x < b by JORDAN6:45; then x in ]. a,+infty.[ & x <> b by Th14; hence thesis by ZFMISC_1:64; end; theorem for A being Subset of R^1, a, b being real number st a <= b & A = {a} \/ [. b,+infty.[ holds A` = ].-infty, a .[ \/ ]. a, b .[ proof let A be Subset of R^1, a, b be real number; assume A1: a <= b & A = {a} \/ [. b,+infty.[; then A` = REAL \ ({a} \/ [. b,+infty.[) by Th13,PRE_TOPC:17 .= (REAL \ [. b,+infty.[) \ {a} by XBOOLE_1:41 .= ].-infty,b.[ \ {a} by Th25; hence thesis by A1,Th93; end; theorem for A being Subset of R^1, a, b being real number st a < b & A = ].-infty, a .[ \/ ]. a, b .[ holds Cl A = ].-infty, b .] proof let A be Subset of R^1, a, b be real number; assume A1: a < b & A = ].-infty, a .[ \/ ]. a, b .[; reconsider B = ].-infty, a .[, C = ]. a, b .[ as Subset of R^1 by TOPMETR:24; A2: Cl C = [. a, b .] by A1,Th36; Cl A = Cl B \/ Cl C by A1,PRE_TOPC:50 .= ].-infty, a .] \/ [. a, b .] by A2,Th77 .= ].-infty, b .] by A1,Th29; hence thesis; end; theorem Th97: for A being Subset of R^1, a, b being real number st a < b & A = ].-infty, a .[ \/ ]. a, b .] holds Cl A = ].-infty, b .] proof let A be Subset of R^1, a, b be real number; assume A1: a < b & A = ].-infty, a .[ \/ ]. a, b .]; reconsider B = ].-infty, a .[, C = ]. a, b .] as Subset of R^1 by TOPMETR:24; A2: Cl C = [. a, b .] by A1,Th59; Cl A = Cl B \/ Cl C by A1,PRE_TOPC:50 .= ].-infty, a .] \/ [. a, b .] by A2,Th77 .= ].-infty, b .] by A1,Th29; hence thesis; end; theorem for A being Subset of R^1, a being real number st A = ].-infty, a .] holds A` = ]. a ,+infty.[ proof let A be Subset of R^1, a be real number; assume A1: A = ].-infty, a .]; A` = REAL \ A by Th13,PRE_TOPC:17 .= ]. a ,+infty.[ by A1,Th23; hence thesis; end; theorem for A being Subset of R^1, a being real number st A = [. a ,+infty.[ holds A` = ].-infty, a .[ proof let A be Subset of R^1, a be real number; assume A1: A = [. a ,+infty.[; A` = REAL \ A by Th13,PRE_TOPC:17 .= ].-infty, a .[ by A1,Th25; hence thesis; end; theorem Th100: for A being Subset of R^1, a, b, c being real number st a < b & b < c & A = ].-infty, a .[ \/ ]. a, b .] \/ IRRAT(b,c) \/ {c} holds Cl A = ].-infty, c .] proof let A be Subset of R^1, a, b, c be real number; assume A1: a < b & b < c & A = ].-infty, a .[ \/ ]. a, b .] \/ IRRAT(b,c) \/ {c}; reconsider B = ].-infty, a .[, C = ]. a, b .], D = IRRAT(b,c), E = {c} as Subset of R^1 by TOPMETR:24; A2: c in E & c in ].-infty, c .] by Th16,TARSKI:def 1; Cl A = Cl (B \/ C \/ D) \/ Cl E by A1,PRE_TOPC:50 .= Cl (B \/ C \/ D) \/ E by Th61 .= Cl (B \/ C) \/ Cl D \/ E by PRE_TOPC:50 .= ].-infty, b .] \/ Cl D \/ E by A1,Th97 .= ].-infty, b .] \/ [. b,c .] \/ E by A1,Th55 .= ].-infty, c .] \/ E by A1,Th29 .= ].-infty, c .] by A2,ZFMISC_1:46; hence thesis; end; theorem for A being Subset of R^1, a, b, c, d being real number st a < b & b < c & A = ].-infty, a .[ \/ ]. a, b .] \/ IRRAT(b,c) \/ {c} \/ {d} holds Cl A = ].-infty, c .] \/ {d} proof let A be Subset of R^1, a, b, c, d be real number; assume A1: a < b & b < c & A = ].-infty, a .[ \/ ]. a, b .] \/ IRRAT(b,c) \/ {c} \/ {d}; reconsider B = ].-infty, a .[, C = ]. a, b .], D = IRRAT(b,c), E = {c}, F = {d} as Subset of R^1 by TOPMETR:24; Cl A = Cl (B \/ C \/ D \/ E) \/ Cl F by A1,PRE_TOPC:50 .= Cl (B \/ C \/ D \/ E) \/ {d} by Th61; hence thesis by A1,Th100; end; theorem for A being Subset of R^1, a, b being real number st a <= b & A = ].-infty, a .] \/ {b} holds A` = ]. a, b .[ \/ ]. b,+infty.[ proof let A be Subset of R^1, a, b be real number; assume A1: a <= b & A = ].-infty, a .] \/ {b}; A` = REAL \ A by Th13,PRE_TOPC:17 .= (REAL \ ].-infty, a .]) \ {b} by A1,XBOOLE_1:41 .= ]. a,+infty.[ \ {b} by Th23 .= ]. a, b .[ \/ ]. b,+infty.[ by A1,Th94; hence thesis; end; theorem for a, b being real number holds [. a,+infty.[ \/ {b} <> REAL proof let a, b be real number; set ab = min (a,b) - 1; A1: ab in REAL by XREAL_0:def 1; A2: min (a,b) <= a & min (a,b) <= b by SQUARE_1:35; A3: ab < min (a,b) by INT_1:26; then ab < a & ab < b by A2,AXIOMS:22; then A4: not ab in [. a,+infty.[ by Th15; not ab in {b} by A2,A3,TARSKI:def 1; hence thesis by A1,A4,XBOOLE_0:def 2; end; theorem for a, b being real number holds ].-infty, a .] \/ {b} <> REAL proof let a, b be real number; set ab = max (a,b) + 1; A1: ab in REAL by XREAL_0:def 1; A2: max (a,b) >= a & max (a,b) >= b by SQUARE_1:46; A3: ab > max (a,b) by REAL_1:69; then ab > a & ab > b by A2,AXIOMS:22; then A4: not ab in ].-infty, a.] by Th16; not ab in {b} by A2,A3,TARSKI:def 1; hence thesis by A1,A4,XBOOLE_0:def 2; end; theorem for TS being TopStruct, A, B being Subset of TS st A <> B holds A` <> B` proof let TS be TopStruct, A, B be Subset of TS; assume A1: A <> B; assume A` = B`; then A = B``; hence thesis by A1; end; theorem for A being Subset of R^1 st REAL = A` holds A = {} proof let A be Subset of R^1; reconsider AB = {}R^1 as Subset of R^1; assume REAL = A`; then [#]R^1 = A` by PRE_TOPC:12,TOPMETR:24; then AB` = A` by PRE_TOPC:27; then AB = A``; hence thesis; end; begin :: Subcontinua of a real line definition cluster I[01] -> arcwise_connected; coherence proof for a, b being Point of I[01] ex f being map of I[01], I[01] st f is continuous & f.0 = a & f.1 = b proof let a, b be Point of I[01]; per cases; suppose A1: a <= b; then reconsider B = [. a, b .] as non empty Subset of I[01] by BORSUK_4:49; 0 <= a & b <= 1 by JORDAN5A:2; then A2: Closed-Interval-TSpace(a,b) = I[01]|B by A1,JGRAPH_5:7; then A3: L[01]((#)(a,b),(a,b)(#)) is continuous map of I[01], I[01]|B by A1,TOPMETR:27,TREAL_1:11; the carrier of I[01]|B c= the carrier of I[01] by BORSUK_1:35; then reconsider g = L[01]((#)(a,b),(a,b)(#)) as Function of the carrier of I[01], the carrier of I[01] by A2,FUNCT_2:9,TOPMETR:27; reconsider g as map of I[01], I[01]; take g; thus g is continuous by A3,TOPMETR:7; 0 = (#)(0,1) by TREAL_1:def 1; hence g.0 = (#)(a,b) by A1,TREAL_1:12 .= a by A1,TREAL_1:def 1; 1 = (0,1)(#) by TREAL_1:def 2; hence g.1 = (a,b)(#) by A1,TREAL_1:12 .= b by A1,TREAL_1:def 2; suppose A4: b <= a; then reconsider B = [. b, a .] as non empty Subset of I[01] by BORSUK_4:49; 0 <= b & a <= 1 by JORDAN5A:2; then A5: Closed-Interval-TSpace(b,a) = I[01]|B by A4,JGRAPH_5:7; then A6: L[01]((#)(b,a),(b,a)(#)) is continuous map of I[01], I[01]|B by A4,TOPMETR:27,TREAL_1:11; the carrier of I[01]|B c= the carrier of I[01] by BORSUK_1:35; then reconsider g = L[01]((#)(b,a),(b,a)(#)) as Function of the carrier of I[01], the carrier of I[01] by A5,FUNCT_2:9,TOPMETR:27; reconsider g as map of I[01], I[01]; A7: g is continuous by A6,TOPMETR:7; 0 = (#)(0,1) by TREAL_1:def 1; then A8: g.0 = (#)(b,a) by A4,TREAL_1:12 .= b by A4,TREAL_1:def 1; 1 = (0,1)(#) by TREAL_1:def 2; then A9: g.1 = (b,a)(#) by A4,TREAL_1:12 .= a by A4,TREAL_1:def 2; then reconsider P = g as Path of b, a by A7,A8,BORSUK_2:def 1; take h = -P; ex t being map of I[01], I[01] st t is continuous & t.0 = a & t.1 = b by A7,A8,A9,Th11; hence thesis by BORSUK_2:def 1; end; hence thesis by BORSUK_2:def 2; end; end; theorem Th107: for X being compact Subset of R^1, X' being Subset of REAL st X' = X holds X' is bounded_above bounded_below proof let X be compact Subset of R^1, X' be Subset of REAL; assume X' = X; then X' is compact by TOPREAL6:81; then X' is bounded by RCOMP_1:28; hence thesis by SEQ_4:def 3; end; theorem Th108: for X being compact Subset of R^1, X' being Subset of REAL, x being real number st x in X' & X' = X holds inf X' <= x & x <= sup X' proof let X be compact Subset of R^1, X' be Subset of REAL, x be real number; assume A1: x in X' & X' = X; then X' is bounded_above bounded_below by Th107; hence thesis by A1,SEQ_4:def 4,def 5; end; theorem Th109: for C being non empty compact connected Subset of R^1, C' being Subset of REAL st C = C' & [. inf C', sup C' .] c= C' holds [. inf C', sup C' .] = C' proof let C be non empty compact connected Subset of R^1, C' be Subset of REAL; assume A1: C = C' & [. inf C', sup C' .] c= C'; assume [. inf C', sup C' .] <> C'; then not C' c= [. inf C', sup C' .] by A1,XBOOLE_0:def 10; then consider c being set such that A2: c in C' & not c in [. inf C', sup C' .] by TARSKI:def 3; reconsider c as real number by A2,XREAL_0:def 1; inf C' <= c & c <= sup C' by A1,A2,Th108; hence thesis by A2,TOPREAL5:1; end; theorem Th110: :: TOPREAL5:9 for A being connected Subset of R^1, a, b, c being real number st a <= b & b <= c & a in A & c in A holds b in A proof let A be connected Subset of R^1, a, b, c be real number; assume A1: a <= b & b <= c & a in A & c in A; per cases by A1,REAL_1:def 5; suppose a = b or b = c; hence thesis by A1; suppose A2: a < b & b < c & a in A & c in A; assume not b in A; then A c= REAL \ {b} by TOPMETR:24,ZFMISC_1:40; then A3: A c= ].-infty,b .[ \/ ]. b,+infty.[ by Th18; reconsider B = ].-infty,b .[, C = ]. b,+infty.[ as Subset of R^1 by TOPMETR:24; A4: B, C are_separated by Th79; now per cases by A3,A4,CONNSP_1:17; suppose A c= B; hence contradiction by A2,Th17; suppose A c= C; hence contradiction by A2,Th14; end; hence thesis; end; theorem Th111: for A being connected Subset of R^1, a, b being real number st a in A & b in A holds [.a,b.] c= A proof let A be connected Subset of R^1, a, b be real number; assume A1: a in A & b in A; let x be set; assume x in [.a,b.]; then x in { y where y is Real : a <= y & y <= b } by RCOMP_1:def 1; then consider z being Real such that A2: z = x & a <= z & z <= b; thus thesis by A1,A2,Th110; end; theorem Th112: for X being non empty compact connected Subset of R^1 holds X is non empty closed-interval Subset of REAL proof let C be non empty compact connected Subset of R^1; reconsider C' = C as non empty Subset of REAL by TOPMETR:24; C is closed by COMPTS_1:16; then A1: C' is closed by TOPREAL6:79; A2: C' is bounded_below bounded_above by Th107; then A3: inf C' in C' by A1,RCOMP_1:31; sup C' in C' by A1,A2,RCOMP_1:30; then [. inf C', sup C' .] c= C' by A3,Th111; then A4: [. inf C', sup C' .] = C' by Th109; C' is bounded by A2,SEQ_4:def 3; then inf C' <= sup C' by SEQ_4:24; hence thesis by A4,INTEGRA1:def 1; end; theorem for A being non empty compact connected Subset of R^1 holds ex a, b being real number st a <= b & A = [. a, b .] proof let C be non empty compact connected Subset of R^1; reconsider C' = C as closed-interval Subset of REAL by Th112; A1: C' = [. inf C', sup C' .] by INTEGRA1:5; inf C' <= sup C' by BORSUK_4:53; then inf C' in C & sup C' in C by A1,RCOMP_1:15; then reconsider p1 = inf C', p2 = sup C' as Point of R^1; take p1, p2; thus p1 <= p2 by BORSUK_4:53; thus thesis by INTEGRA1:5; end; begin :: Sets with Proper Subsets Only definition let TS be TopStruct; let F be Subset-Family of TS; attr F is with_proper_subsets means :Def6: not the carrier of TS in F; end; theorem for TS being TopStruct, F, G being Subset-Family of TS st F is with_proper_subsets & G c= F holds G is with_proper_subsets proof let TS be TopStruct, F, G being Subset-Family of TS; assume A1: F is with_proper_subsets & G c= F; assume not G is with_proper_subsets; then the carrier of TS in G by Def6; hence thesis by A1,Def6; end; definition let TS be non empty TopStruct; cluster with_proper_subsets Subset-Family of TS; existence proof { {} } is Subset-Family of TS by MEASURE1:9; then reconsider F = { {} } as Subset-Family of TS; take F; assume the carrier of TS in F; hence thesis by TARSKI:def 1; end; end; theorem for TS being non empty TopStruct, A, B being with_proper_subsets Subset-Family of TS holds A \/ B is with_proper_subsets proof let TS be non empty TopStruct, A, B be with_proper_subsets Subset-Family of TS; assume the carrier of TS in A \/ B; then the carrier of TS in A or the carrier of TS in B by XBOOLE_0:def 2; hence thesis by Def6; end; definition let T be TopStruct, F be Subset-Family of T; attr F is open means :: TOPS_2:def 1 for P being Subset of T holds P in F implies P is open; attr F is closed means :: TOPS_2:def 2 for P being Subset of T holds P in F implies P is closed; end; definition let T be TopSpace; cluster open closed non empty Subset-Family of T; existence proof reconsider F = {{}T} as Subset-Family of T by Th30; A1: F is open proof let P be Subset of T; assume P in F; hence thesis by TARSKI:def 1; end; F is closed proof let P be Subset of T; assume P in F; hence thesis by TARSKI:def 1; end; hence thesis by A1; end; end;

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