### The Mizar article:

### The Theorem of Weierstrass

**by****Jozef Bialas, and****Yatsuka Nakamura**

- Received July 10, 1995
Copyright (c) 1995 Association of Mizar Users

- MML identifier: WEIERSTR
- [ MML identifier index ]

environ vocabulary METRIC_1, PRE_TOPC, BOOLE, ABSVALUE, ARYTM_1, SETFAM_1, FINSEQ_1, FINSET_1, TOPMETR, RELAT_1, TARSKI, FUNCT_1, SUBSET_1, ORDINAL2, COMPTS_1, EUCLID, LATTICES, PCOMPS_1, SEQ_2, SEQ_1, SEQM_3, SEQ_4, WEIERSTR, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2, ORDINAL1, NUMBERS, XREAL_0, REAL_1, NAT_1, TOPMETR, METRIC_1, COMPTS_1, SEQ_1, SEQ_2, SEQM_3, SEQ_4, ABSVALUE, FINSEQ_1, FINSET_1, STRUCT_0, PRE_TOPC, TOPS_2, RCOMP_1, EUCLID, PCOMPS_1; constructors REAL_1, NAT_1, TOPMETR, COMPTS_1, SEQ_2, SEQM_3, SEQ_4, ABSVALUE, TOPS_2, RCOMP_1, EUCLID; clusters STRUCT_0, TOPMETR, PRE_TOPC, PCOMPS_1, EUCLID, XREAL_0, METRIC_1, RELSET_1, ARYTM_3, FINSET_1, MEMBERED, NUMBERS, ORDINAL2; requirements REAL, SUBSET, BOOLE, NUMERALS, ARITHM; definitions STRUCT_0, TARSKI; theorems TARSKI, AXIOMS, FUNCT_1, FUNCT_2, NAT_1, FINSEQ_1, ZFMISC_1, REAL_1, SEQ_2, SEQ_4, ABSVALUE, COMPTS_1, PRE_TOPC, RELAT_1, TOPS_2, RCOMP_1, TOPMETR, PCOMPS_1, METRIC_1, METRIC_6, SETFAM_1, XBOOLE_0, XBOOLE_1, XREAL_0, XCMPLX_0, XCMPLX_1; schemes NAT_1, PRE_TOPC, FUNCT_2; begin theorem Th1: for M being MetrSpace, x1,x2 being Point of M, r1,r2 being Real ex x being Point of M, r being Real st Ball(x1,r1) \/ Ball(x2,r2) c= Ball(x,r) proof let M be MetrSpace; let x1,x2 be Point of M; let r1,r2 be Real; reconsider x = x1 as Point of M; reconsider r = abs(r1) + abs(r2) + dist(x1,x2) as Real; A1: for a being set holds a in Ball(x1,r1) \/ Ball(x2,r2) implies a in Ball(x,r) proof let a be set; assume A2: a in (Ball(x1,r1) \/ Ball(x2,r2)); then reconsider a as Point of M; now per cases by A2,XBOOLE_0:def 2; case A3:a in Ball(x1,r1); A4:M is non empty proof thus the carrier of M is non empty by A3; end; A5:dist(x,a) < r1 by A3,METRIC_1:12; r1 <= abs(r1) & 0 <= abs(r2) by ABSVALUE:5,11; then A6: r1 + 0 <= abs(r1) + abs(r2) by REAL_1:55; 0 <= dist(x1,x2) by METRIC_1:5; then r1 + 0 <= abs(r1) + abs(r2) + dist(x1,x2) by A6,REAL_1:55; then dist(x,a) - r < r1 - r1 by A5,REAL_1:92; then dist(x,a) - r < r1 + (-r1) by XCMPLX_0:def 8; then dist(x,a) - r < 0 & r <= r by XCMPLX_0:def 6; then dist(x,a) - r + r < 0 + r by REAL_1:67; then dist(x,a) + (- r) + r < r by XCMPLX_0:def 8; then dist(x,a) + ((- r) + r) < r by XCMPLX_1:1; then dist(x,a) + 0 < r by XCMPLX_0:def 6; hence thesis by A4,METRIC_1:12; case A7:a in Ball(x2,r2); A8:M is non empty proof thus the carrier of M is non empty by A7; end; A9:dist(x,a) <= dist(x1,x2) + dist(x2,a) by METRIC_1:4; A10:dist(x2,a) < r2 by A7,METRIC_1:12; r2 <= abs(r2) by ABSVALUE:11; then dist(x2,a) - abs(r2) < r2 - r2 by A10,REAL_1:92; then dist(x2,a) - abs(r2) < r2 + (-r2) by XCMPLX_0:def 8; then dist(x2,a) - abs(r2) < 0 & abs(r2) <= abs(r2) by XCMPLX_0:def 6; then dist(x2,a) - abs(r2) + abs(r2) < 0 + abs(r2) by REAL_1:67; then dist(x2,a) + (- abs(r2)) + abs(r2) < abs(r2) by XCMPLX_0:def 8; then dist(x2,a) + ((- abs(r2)) + abs(r2)) < abs(r2) by XCMPLX_1:1; then dist(x2,a) + 0 < abs(r2) by XCMPLX_0:def 6; then dist(x,a) - abs(r2) < dist(x1,x2) + dist(x2,a) - dist(x2,a) by A9,REAL_1:92; then dist(x,a) - abs(r2) < dist(x1,x2) + dist(x2,a) + (-dist(x2,a)) by XCMPLX_0:def 8; then dist(x,a) - abs(r2) < dist(x1,x2) + (dist(x2,a) + (-dist(x2,a))) by XCMPLX_1:1; then A11: dist(x,a) - abs(r2) < dist(x1,x2) + 0 by XCMPLX_0:def 6; 0 <= abs(r1) by ABSVALUE:5; then dist(x,a) - abs(r2) - abs(r1) < dist(x1,x2) - 0 by A11,REAL_1:92 ; then dist(x,a) - (abs(r1) + abs(r2)) < dist(x1,x2) by XCMPLX_1:36; then dist(x,a) - (abs(r1) + abs(r2)) + (abs(r1) + abs(r2)) < abs(r1) + abs(r2) + dist(x1,x2) by REAL_1:67; then dist(x,a) + (-(abs(r1) + abs(r2))) + (abs(r1) + abs(r2)) < abs(r1) + abs(r2) + dist(x1,x2) by XCMPLX_0:def 8; then dist(x,a) + (-(abs(r1) + abs(r2)) + (abs(r1) + abs(r2))) < abs(r1) + abs(r2) + dist(x1,x2) by XCMPLX_1:1; then dist(x,a) + 0 < abs(r1) + abs(r2) + dist(x1,x2) by XCMPLX_0:def 6; hence thesis by A8,METRIC_1:12; end; hence thesis; end; take x; take r; thus thesis by A1,TARSKI:def 3; end; theorem Th2: for M being MetrSpace, n being Nat, F being Subset-Family of M, p being FinSequence st (F is finite & F is_ball-family & rng p = F & dom p = Seg(n+1)) holds ex G being Subset-Family of M st (G is finite & G is_ball-family & ex q being FinSequence st (rng q = G & dom q = Seg(n)) & ex x being Point of M st ex r being Real st (union F c= union G \/ Ball(x,r))) proof let M be MetrSpace; let n be Nat; let F be Subset-Family of M; let p be FinSequence; assume A1:F is finite & F is_ball-family & rng p = F & dom p = Seg(n+1); reconsider q = p|(Seg(n)) as FinSequence by FINSEQ_1:19; len p = n+1 by A1,FINSEQ_1:def 3; then n <= len p by NAT_1:29; then A2:len q = n & dom q = Seg(n) by FINSEQ_1:21; then A3:dom q \/ {n+1} = dom p by A1,FINSEQ_1:11; A4:dom q c= dom p & rng q c= rng p by FUNCT_1:76; then rng q c= bool the carrier of M by A1,XBOOLE_1:1; then reconsider G=rng q as Subset-Family of M by SETFAM_1:def 7; reconsider G as Subset-Family of M; A5:G is_ball-family proof for x being set holds x in G implies ex y being Point of M st ex r being Real st x = Ball(y,r) by A1,A4,TOPMETR:def 4; hence thesis by TOPMETR:def 4; end; n+1 in dom p by A1,FINSEQ_1:6; then p.(n+1) in F by A1,FUNCT_1:def 5; then consider x being Point of M such that A6:ex r being Real st p.(n+1) = Ball(x,r) by A1,TOPMETR:def 4; consider r being Real such that A7:p.(n+1) = Ball(x,r) by A6; A8:ex x being Point of M st ex r being Real st (union F c= union G \/ Ball(x,r)) proof A9:union F c= union G \/ Ball(x,r) proof let t be set; assume t in union F; then consider A being set such that A10:t in A & A in F by TARSKI:def 4; consider s being set such that A11:s in dom p & A = p.s by A1,A10,FUNCT_1:def 5; now per cases by A3,A11,XBOOLE_0:def 2; case A12:s in dom q; then A13:q.s in G by FUNCT_1:def 5; q.s = A by A11,A12,FUNCT_1:70; then A14:t in union G by A10,A13,TARSKI:def 4; union G c= union G \/ Ball(x,r) by XBOOLE_1:7; hence thesis by A14; case s in {n+1}; then p.s = Ball(x,r) by A7,TARSKI:def 1; hence thesis by A10,A11,XBOOLE_0:def 2; end; hence thesis; end; take x; take r; thus thesis by A9; end; take G; thus thesis by A2,A5,A8,FINSEQ_1:73; end; theorem Th3: for M being MetrSpace, F being Subset-Family of M st (F is finite & F is_ball-family) holds ex x being Point of M, r being Real st union F c= Ball(x,r) proof let M be MetrSpace; let F be Subset-Family of M; assume A1:F is finite & F is_ball-family; A2:for F being Subset-Family of M st (F is finite & F is_ball-family) holds for n being Nat holds for p being FinSequence st (rng p = F & dom p = Seg n) holds (ex x being Point of M , r being Real st union F c= Ball(x,r)) proof defpred P[Nat] means for F being Subset-Family of M st (F is finite & F is_ball-family) holds for n being Nat st n = $1 holds for p being FinSequence st (rng p = F & dom p = Seg(n)) holds (ex x being Point of M st ex r being Real st union F c= Ball(x,r)); A3: P[0] proof let F be Subset-Family of M; assume F is finite & F is_ball-family; let n be Nat; assume A4:n = 0; for p being FinSequence st (rng p = F & dom p = Seg(n)) holds (ex x being Point of M st ex r being Real st union F c= Ball(x,r)) proof let p be FinSequence; assume rng p = F & dom p = Seg(n); then A5:union F = {} by A4,FINSEQ_1:4,RELAT_1:65,ZFMISC_1:2; consider x being Point of M; consider r being Real; take x; take r; thus thesis by A5,XBOOLE_1:2; end; hence thesis; end; A6:for k being Nat st P[k] holds P[k + 1] proof let k be Nat; assume A7:P[k]; let F be Subset-Family of M; assume A8:F is finite & F is_ball-family; let n be Nat; assume A9:n = k+1; let p be FinSequence; assume rng p = F & dom p = Seg(n); then consider F1 being Subset-Family of M such that A10:F1 is finite & F1 is_ball-family & ex p1 being FinSequence st (rng p1 = F1 & dom p1 = Seg(k)) & ex x2 being Point of M st ex r2 being Real st (union F c= union F1 \/ Ball(x2,r2)) by A8,A9,Th2; consider x2 being Point of M such that A11:ex r2 being Real st (union F c= union F1 \/ Ball(x2,r2)) by A10; consider r2 being Real such that A12:union F c= union F1 \/ Ball(x2,r2) by A11; consider x1 being Point of M such that A13:ex r being Real st union F1 c= Ball(x1,r) by A7,A10; consider r1 being Real such that A14:union F1 c= Ball(x1,r1) by A13; A15:union F1 \/ Ball(x2,r2) c= Ball(x1,r1) \/ Ball(x2,r2) by A14,XBOOLE_1:9; consider x being Point of M such that A16:ex r being Real st Ball(x1,r1) \/ Ball(x2,r2) c= Ball(x,r) by Th1; consider r being Real such that A17:Ball(x1,r1) \/ Ball(x2,r2) c= Ball(x,r) by A16; take x; take r; union F c= Ball(x1,r1) \/ Ball(x2,r2) by A12,A15,XBOOLE_1:1; hence thesis by A17,XBOOLE_1:1; end; for n being Nat holds P[n] from Ind(A3,A6); hence thesis; end; consider p being FinSequence such that A18:rng p = F by A1,FINSEQ_1:73; consider n being Nat such that A19:dom p = Seg n by FINSEQ_1:def 2; thus thesis by A1,A2,A18,A19; end; definition let T,S be non empty TopSpace; let f be map of T,S; let G be Subset-Family of S; func f"G -> Subset-Family of T means :Def1:for A being Subset of T holds A in it iff ex B being Subset of S st (B in G & A = f"B); existence proof defpred P[Subset of T] means ex B being Subset of S st B in G & $1 = f"B; ex R being Subset-Family of T st for A being Subset of T holds A in R iff P[A] from SubFamExS; hence thesis; end; uniqueness proof let R1,R2 be Subset-Family of T such that A1:for A being Subset of T holds A in R1 iff ex B being Subset of S st (B in G & A = f"B) and A2:for A being Subset of T holds A in R2 iff ex B being Subset of S st (B in G & A = f"B); for x being set holds (x in R1 iff x in R2) proof let x be set; thus x in R1 implies x in R2 proof assume A3:x in R1; then reconsider x as Subset of T; ex B being Subset of S st (B in G & x = f"B) by A1,A3; hence thesis by A2; end; assume A4:x in R2; then reconsider x as Subset of T; ex B being Subset of S st (B in G & x = f"B) by A2,A4; hence thesis by A1; end; hence thesis by TARSKI:2; end; end; theorem for T,S being non empty TopSpace, f being map of T,S, A,B being Subset-Family of S st A c= B holds f"A c= f"B proof let T,S be non empty TopSpace; let f be map of T,S; let A,B be Subset-Family of S; assume A1:A c= B; let x be set; assume A2:x in f"A; then reconsider x as Subset of T; ex C being Subset of S st C in B & x = f"C proof consider C being Subset of S such that A3:C in A & x = f"C by A2,Def1; take C; thus thesis by A1,A3; end; hence thesis by Def1; end; theorem Th5: for T,S being non empty TopSpace, f being map of T,S, B being Subset-Family of S st f is continuous & B is open holds f"B is open proof let T,S be non empty TopSpace; let f be map of T,S; let B be Subset-Family of S; assume A1:f is continuous & B is open; for P being Subset of T holds P in f"B implies P is open proof let P be Subset of T; assume A2:P in f"(B); thus P is open proof consider C being Subset of S such that A3:C in B & P = f"C by A2,Def1; reconsider C as Subset of S; C is open by A1,A3,TOPS_2:def 1; hence thesis by A1,A3,TOPS_2:55; end; end; hence thesis by TOPS_2:def 1; end; definition let T,S be non empty TopSpace; let f be map of T,S; let G be Subset-Family of T; func f.:G -> Subset-Family of S means :Def2:for A being Subset of S holds A in it iff ex B being Subset of T st (B in G & A = f.:B); existence proof thus ex R being Subset-Family of S st for A being Subset of S holds A in R iff ex B being Subset of T st (B in G & A = f.:B) proof defpred P[Subset of S] means ex B being Subset of T st B in G & $1 = f.:B; ex R being Subset-Family of S st for A being Subset of S holds A in R iff P[A] from SubFamExS; hence thesis; end; end; uniqueness proof let R1,R2 be Subset-Family of S such that A1:for A being Subset of S holds A in R1 iff ex B being Subset of T st (B in G & A = f.:B) and A2:for A being Subset of S holds A in R2 iff ex B being Subset of T st (B in G & A = f.:B); for x being set holds (x in R1 iff x in R2) proof let x be set; thus x in R1 implies x in R2 proof assume A3:x in R1; then reconsider x as Subset of S; ex B being Subset of T st (B in G & x = f.:B) by A1,A3; hence thesis by A2; end; assume A4:x in R2; then reconsider x as Subset of S; ex B being Subset of T st (B in G & x = f.:B) by A2,A4; hence thesis by A1; end; hence thesis by TARSKI:2; end; end; theorem for T,S being non empty TopSpace, f being map of T,S, A,B being Subset-Family of T holds A c= B implies f.:A c= f.:B proof let T,S be non empty TopSpace; let f be map of T,S; let A,B be Subset-Family of T; assume A1:A c= B; let x be set; assume A2:x in f.:A; then reconsider x as Subset of S; ex C being Subset of T st (C in B & x = f.:C) proof consider C being Subset of T such that A3:C in A & x = f.:C by A2,Def2; take C; thus thesis by A1,A3; end; hence thesis by Def2; end; theorem for T,S being non empty TopSpace, f being map of T,S, B being Subset-Family of S, P being Subset of S st f.:(f"B) is_a_cover_of P holds B is_a_cover_of P proof let T,S be non empty TopSpace; let f be map of T,S; let B be Subset-Family of S; let P be Subset of S; assume f.:(f"B) is_a_cover_of P; then A1:P c= union (f.:(f"B)) by COMPTS_1:def 1; P c= union B proof let x be set; assume x in P; then consider Y being set such that A2:x in Y & Y in f.:(f"(B)) by A1,TARSKI:def 4; ex Z being set st (x in Z & Z in B) proof reconsider Y as Subset of S by A2; consider Y1 being Subset of T such that A3:Y1 in f"(B) & Y = f.:Y1 by A2,Def2; consider Y2 being Subset of S such that A4:Y2 in B & Y1 = f"(Y2) by A3,Def1; A5: f.:(f"Y2) c= Y2 by FUNCT_1:145; reconsider Y2 as set; take Y2; thus thesis by A2,A3,A4,A5; end; hence thesis by TARSKI:def 4; end; hence thesis by COMPTS_1:def 1; end; theorem for T,S being non empty TopSpace, f being map of T,S, B being Subset-Family of T, P being Subset of T st B is_a_cover_of P holds f"(f.:B) is_a_cover_of P proof let T,S be non empty TopSpace; let f be map of T,S; let B be Subset-Family of T; let P be Subset of T; assume B is_a_cover_of P; then A1:P c= union B by COMPTS_1:def 1; P c= union(f"(f.:B)) proof let x be set; assume x in P; then consider Y being set such that A2:x in Y & Y in B by A1,TARSKI:def 4; ex Z being set st (x in Z & Z in f"(f.:(B))) proof A3: B c= bool [#](T) by TOPS_2:1; reconsider Y as Subset of T by A2; set Z = f"(f.:Y); dom f = [#](T) by TOPS_2:51; then A4: Y c= f"(f.:Y) by A2,A3,FUNCT_1:146; A5: f.:(Y) in f.:B & f"(f.:Y) = f"(f.:Y) by A2,Def2; take Z; thus thesis by A2,A4,A5,Def1; end; hence thesis by TARSKI:def 4; end; hence thesis by COMPTS_1:def 1; end; theorem for T,S being non empty TopSpace, f being map of T,S, Q being Subset-Family of S holds union(f.:(f"(Q))) c= union Q proof let T,S be non empty TopSpace; let f be map of T,S; let Q be Subset-Family of S; let x be set; thus x in union(f.:(f"(Q))) implies x in union Q proof assume x in union(f.:(f"Q)); then consider A being set such that A1:x in A & A in f.:(f"Q) by TARSKI:def 4; reconsider A as Subset of S by A1; consider A1 being Subset of T such that A2:A1 in f"Q & A = f.:A1 by A1,Def2; consider A2 being Subset of S such that A3:A2 in Q & A1 = f"A2 by A2,Def1; f.:(f"A2) c= A2 by FUNCT_1:145; hence thesis by A1,A2,A3,TARSKI:def 4; end; end; theorem for T,S being non empty TopSpace, f being map of T,S, P being Subset-Family of T holds union P c= union(f"(f.:P)) proof let T,S be non empty TopSpace; let f be map of T,S; let P be Subset-Family of T; let x be set; assume x in union P; then consider A being set such that A1:x in A & A in P by TARSKI:def 4; P c= bool [#](T) by TOPS_2:1; then A2:A c= [#](T) by A1; reconsider A as Subset of T by A1; reconsider A1 = f.:A as Subset of S; reconsider A2 = f"A1 as Subset of T; A c= dom f by A2,TOPS_2:51; then A3: A c= A2 by FUNCT_1:146; A1 in f.:P by A1,Def2; then A2 in f"(f.:P) by Def1; hence thesis by A1,A3,TARSKI:def 4; end; theorem for T,S being non empty TopSpace holds for f being map of T,S holds for Q being Subset-Family of S holds Q is finite implies f"Q is finite proof let T,S be non empty TopSpace; let f be map of T,S; let Q be Subset-Family of S; assume Q is finite; then consider s being FinSequence such that A1:rng s = Q by FINSEQ_1:73; defpred EF[Element of bool [#](S),Element of bool [#](T)] means for s,t being set holds ($1 = s & $2 = t implies t = f"s); A2:for x being Element of bool [#](S) ex y being Element of bool [#](T) st EF[x,y] proof let x be Element of bool [#](S); reconsider x as set; set y = f"x; for z being set holds z in f"x implies z in [#](T) proof let z be set; assume z in f"x; then z in dom f & f.z in x by FUNCT_1:def 13; hence thesis by TOPS_2:51; end; then reconsider y as Element of bool [#](T) by TARSKI:def 3; take y; thus thesis; end; consider F being Function of bool [#](S),bool [#](T) such that A3:for x being Element of bool [#](S) holds EF[x,F.x] from FuncExD(A2); Q c= bool [#](S) & dom F = bool [#](S) by FUNCT_2:def 1,TOPS_2:1; then reconsider q = F*s as FinSequence by A1,FINSEQ_1:20; A4: F.:Q = f"Q proof for x being set holds x in F.:Q iff x in f"Q proof let x be set; thus x in F.:Q implies x in f"Q proof assume x in F.:Q; then consider y being set such that A5:y in dom F & y in Q & x = F.y by FUNCT_1:def 12; A6:dom F = bool [#](S) by FUNCT_2:def 1; reconsider y as Subset of S by A5; F.y = f"y by A3,A5,A6; hence thesis by A5,Def1; end; assume A7:x in f"Q; then reconsider x as Subset of T; consider y being Subset of S such that A8:y in Q & x = f"y by A7,Def1; A9: Q c= bool [#](S) by TOPS_2:1; then A10:x = F.y by A3,A8; dom F = bool [#](S) by FUNCT_2:def 1; hence thesis by A8,A9,A10,FUNCT_1:def 12; end; hence thesis by TARSKI:2; end; ex q being FinSequence st rng q = f"Q proof take q; thus thesis by A1,A4,RELAT_1:160; end; hence thesis by FINSEQ_1:73; end; theorem for T,S being non empty TopSpace holds for f being map of T,S holds for P being Subset-Family of T holds P is finite implies f.:P is finite proof let T,S be non empty TopSpace; let f be map of T,S; let P be Subset-Family of T; assume P is finite; then consider s being FinSequence such that A1:rng s = P by FINSEQ_1:73; defpred EF[Element of bool [#](T),Element of bool [#](S)] means for s,t being set holds ($1 = s & $2 = t implies t = f.:s); A2:for x being Element of bool [#](T) ex y being Element of bool [#](S) st EF[x,y] proof let x be Element of bool [#](T); reconsider x as set; set y = f.:x; f.:x c= [#](S) proof let z be set; assume z in f.:x; then consider w being set such that A3:w in dom f & w in x & z = f.w by FUNCT_1:def 12; [#](T) = the carrier of T & [#](S) = the carrier of S by PRE_TOPC:12; then reconsider f as Function of [#](T),[#](S); f.w in [#](S) by A3,FUNCT_2:7; hence thesis by A3; end; then reconsider y as Element of bool [#](S); take y; thus thesis; end; consider F being Function of bool [#](T),bool [#](S) such that A4:for x being Element of bool [#](T) holds EF[x,F.x] from FuncExD(A2); P c= bool [#](T) & dom F = bool [#](T) by FUNCT_2:def 1,TOPS_2:1; then reconsider q = F*s as FinSequence by A1,FINSEQ_1:20; A5: F.:P = f.:P proof for x being set holds x in F.:P iff x in f.:P proof let x be set; thus x in F.:P implies x in f.:P proof assume x in F.:P; then consider y being set such that A6:y in dom F & y in P & x = F.y by FUNCT_1:def 12; A7:dom F = bool [#](T) by FUNCT_2:def 1; reconsider y as Subset of T by A6; F.y = f.:y by A4,A6,A7; hence thesis by A6,Def2; end; thus x in f.:P implies x in F.:P proof assume A8:x in f.:P; then reconsider x as Subset of S; consider y being Subset of T such that A9:y in P & x = f.:y by A8,Def2; A10: P c= bool [#](T) by TOPS_2:1; then A11:x = F.y by A4,A9; dom F = bool [#](T) by FUNCT_2:def 1; hence thesis by A9,A10,A11,FUNCT_1:def 12; end; end; hence thesis by TARSKI:2; end; ex q being FinSequence st rng q = f.:P proof take q; thus thesis by A1,A5,RELAT_1:160; end; hence thesis by FINSEQ_1:73; end; theorem Th13: for T,S being non empty TopSpace holds for f being map of T,S holds for P being Subset of T holds for F being Subset-Family of S holds (ex B being Subset-Family of T st (B c= f"F & B is_a_cover_of P & B is finite)) implies (ex G being Subset-Family of S st (G c= F & G is_a_cover_of f.:P & G is finite)) proof let T,S be non empty TopSpace; let f be map of T,S; let P be Subset of T; let F be Subset-Family of S; given B being Subset-Family of T such that A1:B c= f"F & B is_a_cover_of P & B is finite; A2:P c= union B by A1,COMPTS_1:def 1; now per cases; case A3:P <> {}; thus ex G being Subset-Family of S st (G c= F & G is_a_cover_of f.:P & G is finite) proof B <> {} by A2,A3,XBOOLE_1:3,ZFMISC_1:2; then consider D being non empty set such that A4: D = B; defpred P0[Element of D,Element of (bool [#](S))] means for x being Element of D st $1 = x holds for y being Element of (bool [#](S)) st $2 = y holds (y in F & x = f"y); A5:for x being Element of D ex y being Element of bool [#](S) st P0[x,y] proof let x be Element of D; A6:x in B by A4; then reconsider x as Subset of T; consider y being Subset of S such that A7:y in F & x = f"y by A1,A6,Def1; reconsider y as Element of bool [#](S) by PRE_TOPC:12; take y; thus thesis by A7; end; consider F0 being Function of D,bool [#](S) such that A8:for x being Element of D holds P0[x,F0.x] from FuncExD(A5); A9: for x being Element of D holds (F0.x in F & x = f"(F0.x)) by A8; reconsider F0 as Function of B,bool [#](S) by A4; consider s being FinSequence such that A10:rng s = B by A1,FINSEQ_1:73; A11:dom F0 = B by FUNCT_2:def 1; then reconsider q = F0*s as FinSequence by A10,FINSEQ_1:20; set G = rng q; A12:G c= F proof let x be set; assume x in G; then consider y being set such that A13:y in dom q & x = q.y by FUNCT_1:def 5; A14:s.y in B by A11,A13,FUNCT_1:21; x = F0.(s.y) by A13,FUNCT_1:22; hence thesis by A4,A9,A14; end; then G c= bool the carrier of S by XBOOLE_1:1; then reconsider G as Subset-Family of S by SETFAM_1:def 7; reconsider G as Subset-Family of S; take G; G is_a_cover_of f.:P proof f.:P c= union G proof for x being set holds x in f.:P implies x in union G proof let x be set; assume A15:x in f.:P; ex A being set st x in A & A in G proof consider y being set such that A16:y in dom f & y in P & x = f.y by A15,FUNCT_1:def 12; consider C being set such that A17:y in C & C in B by A2,A16,TARSKI:def 4; F0.C in F & C = f"(F0.C) by A4,A9,A17; then A18:x in f.:(f"(F0.C)) by A16,A17,FUNCT_1:def 12; A19: f.:(f"(F0.C)) c= F0.C by FUNCT_1:145; A20:G = rng F0 by A10,A11,RELAT_1:47; set A = F0.C; take A; thus thesis by A17,A18,A19,A20,FUNCT_2:6; end; hence x in union G by TARSKI:def 4; end; hence thesis by TARSKI:def 3; end; hence thesis by COMPTS_1:def 1; end; hence thesis by A12,FINSEQ_1:73; end; hence thesis; case A21:P = {}; ex G being Subset-Family of S st (G c= F & G is_a_cover_of f.:P & G is finite) proof set G = {}; A22:f.:P = {} proof assume f.:P <> {}; then consider x being set such that A23:x in f.:P by XBOOLE_0:def 1; consider z being set such that A24:z in dom f & z in P & x = f.z by A23,FUNCT_1:def 12; thus contradiction by A21,A24; end; G c= bool the carrier of S by XBOOLE_1:2; then reconsider G as Subset-Family of S by SETFAM_1:def 7; reconsider G as Subset-Family of S; take G; thus thesis by A22,COMPTS_1:def 1,XBOOLE_1:2,ZFMISC_1:2; end; hence thesis; end; hence thesis; end; begin :: :: The Weierstrass` theorem :: theorem Th14: for T,S being non empty TopSpace holds for f being map of T,S holds for P being Subset of T holds (P is compact & f is continuous) implies f.:P is compact proof let T,S be non empty TopSpace; let f be map of T,S; let P be Subset of T; assume A1:P is compact & f is continuous; P is Subset of [#]T by PRE_TOPC:12; then P c= [#]T; then A2:P c= dom f by TOPS_2:51; for F0 being Subset-Family of S st (F0 is_a_cover_of f.:P & F0 is open) ex G being Subset-Family of S st (G c= F0 & G is_a_cover_of f.:P & G is finite) proof let F0 be Subset-Family of S; assume A3:F0 is_a_cover_of f.:P & F0 is open; then A4:f.:P c= union F0 by COMPTS_1:def 1; set B0 = f"F0; A5:B0 is open by A1,A3,Th5; P c= union B0 proof let x be set; thus x in P implies x in union B0 proof assume A6:x in P; then A7:f.x in f.:P by A2,FUNCT_1:def 12; reconsider x as Point of T by A6; A8:f.x in union F0 by A4,A7; A9:f"{f.x} c= f"(union F0) proof let y be set; assume y in f"{f.x}; then y in dom f & f.y in {f.x} by FUNCT_1:def 13; then y in dom f & f.y in union F0 by A8,TARSKI:def 1; hence thesis by FUNCT_1:def 13; end; A10:f"(union F0) c= union(f"F0) proof let y be set; assume A11:y in f"(union F0); thus y in union(f"F0) proof A12:y in dom f & f.y in union F0 by A11,FUNCT_1:def 13; then consider Q being set such that A13:f.y in Q & Q in F0 by TARSKI:def 4; ex Z being set st y in Z & Z in f"F0 proof set Z = f"Q; take Z; thus thesis by A12,A13,Def1,FUNCT_1:def 13; end; hence thesis by TARSKI:def 4; end; end; x in dom f & f.x in {f.x} by A2,A6,TARSKI:def 1; then x in f"{f.x} by FUNCT_1:def 13; then x in f"(union F0) by A9; hence thesis by A10; end; end; then B0 is_a_cover_of P by COMPTS_1:def 1; then consider B being Subset-Family of T such that A14:B c= B0 & B is_a_cover_of P & B is finite by A1,A5,COMPTS_1:def 7; consider G being Subset-Family of S such that A15:G c= F0 & G is_a_cover_of f.:P & G is finite by A14,Th13; take G; thus thesis by A15; end; hence thesis by COMPTS_1:def 7; end; theorem for T being non empty TopSpace holds for f being map of T,R^1 holds for P being Subset of T holds P is compact & f is continuous implies f.:P is compact by Th14; theorem for f being map of TOP-REAL 2,R^1 holds for P being Subset of TOP-REAL 2 holds P is compact & f is continuous implies f.:P is compact by Th14; definition let P be Subset of R^1; func [#](P) -> Subset of REAL equals :Def3: P; correctness by TOPMETR:24; end; theorem Th17: for P being Subset of R^1 holds P is compact implies [#](P) is bounded proof let P be Subset of R^1; assume A1:P is compact; thus [#](P) is bounded proof now per cases; case [#](P) <> {}; set r0 = 1; defpred P[Subset of R^1] means ex x being Point of RealSpace st x in [#](P) & $1 = Ball(x,r0); consider R being Subset-Family of R^1 such that A2:for A being Subset of R^1 holds A in R iff P[A] from SubFamExS; A3:R is open proof for A being Subset of R^1 holds A in R implies A is open proof let A be Subset of R^1; assume A in R; then consider x being Point of RealSpace such that A4:x in [#](P) & A = Ball(x,r0) by A2; thus thesis by A4,TOPMETR:21,def 7; end; hence thesis by TOPS_2:def 1; end; [#](P) c= union R proof for x being set holds x in [#](P) implies x in union R proof let x be set; assume A5:x in [#](P); then reconsider x as Point of RealSpace by METRIC_1:def 14; consider A being Subset of RealSpace such that A6:A = Ball(x,r0); R^1 = TopStruct (#the carrier of RealSpace, Family_open_set(RealSpace)#) by PCOMPS_1:def 6,TOPMETR:def 7; then reconsider A as Subset of R^1; ex A being set st x in A & A in R proof A7: dist(x,x) = 0 by METRIC_1:1; take A; thus thesis by A2,A5,A6,A7,METRIC_1:12; end; hence thesis by TARSKI:def 4; end; hence thesis by TARSKI:def 3; end; then P c= union R by Def3; then R is_a_cover_of P by COMPTS_1:def 1; then consider R0 being Subset-Family of R^1 such that A8:R0 c= R & R0 is_a_cover_of P & R0 is finite by A1,A3,COMPTS_1:def 7; P c= union R0 by A8,COMPTS_1:def 1; then A9:[#](P) c= union R0 by Def3; A10:for A being set holds A in R0 implies ex x being Point of RealSpace,r being Real st A = Ball(x,r) proof let A be set; assume A11: A in R0; then reconsider A as Subset of R^1; consider x being Point of RealSpace such that A12:x in [#](P) & A = Ball(x,r0) by A2,A8,A11; take x; take r0; thus thesis by A12; end; R^1 = TopStruct (#the carrier of RealSpace, Family_open_set(RealSpace)#) by PCOMPS_1:def 6,TOPMETR:def 7; then reconsider R0 as Subset-Family of RealSpace; R0 is_ball-family by A10,TOPMETR:def 4; then consider x1 being Point of RealSpace such that A13:ex r1 being Real st union R0 c= Ball(x1,r1) by A8,Th3; consider r1 being Real such that A14:union R0 c= Ball(x1,r1) by A13; A15:[#](P) c= Ball(x1,r1) by A9,A14,XBOOLE_1:1; reconsider x1 as Real by METRIC_1:def 14; A16:for p being Real holds p in [#](P) implies (x1 - r1 <= p & p <= x1 + r1) proof let p be Real; assume A17:p in [#](P); reconsider a=x1,b=p as Element of RealSpace by METRIC_1:def 14; dist(a,b) < r1 by A15,A17,METRIC_1:12; then abs(x1-p) < r1 by TOPMETR:15; then -r1 <= x1 - p & x1 - p <= r1 by ABSVALUE:12; then -r1 + p <= x1 & x1 <= p + r1 by REAL_1:84,86; then p <= x1 -(-r1) & x1 - r1 <= p by REAL_1:84,86; then p <= x1 + (-(-r1)) & x1 - r1 <= p by XCMPLX_0:def 8; hence thesis; end; A18:[#](P) is bounded_above proof ex p being real number st for r being real number st r in [#](P) holds r <= p proof take x1 + r1; thus thesis by A16; end; hence thesis by SEQ_4:def 1; end; [#](P) is bounded_below proof ex p being real number st for r being real number st r in [#](P) holds p <= r proof take x1 - r1; thus thesis by A16; end; hence thesis by SEQ_4:def 2; end; hence thesis by A18,SEQ_4:def 3; case A19:[#](P) = {}; A20:[#](P) is bounded_above proof ex p being real number st for r being real number st r in [#](P) holds r <= p proof take 0; thus thesis by A19; end; hence thesis by SEQ_4:def 1; end; [#](P) is bounded_below proof ex p being real number st for r being real number st r in [#](P) holds p <= r proof take 0; thus thesis by A19; end; hence thesis by SEQ_4:def 2; end; hence thesis by A20,SEQ_4:def 3; end; hence thesis; end; thus thesis; end; theorem Th18: for P being Subset of R^1 holds P is compact implies [#](P) is closed proof let P be Subset of R^1; assume A1:P is compact; now per cases; case A2:[#](P) <> {}; A3:R^1 is_T2 by PCOMPS_1:38,TOPMETR:def 7; for s1 being Real_Sequence st (rng s1) c= [#](P) & s1 is convergent holds lim s1 in [#](P) proof let s1 be Real_Sequence; assume A4: (rng s1) c= [#](P) & s1 is convergent; then A5:(rng s1) c= P & s1 is convergent by Def3; set x = lim s1; reconsider x as Point of R^1 by TOPMETR:24; thus lim s1 in [#](P) proof assume not lim s1 in [#](P); then A6: not lim s1 in P by Def3; P <> {} by A2,Def3; then consider Otx,OtP being Subset of R^1 such that A7:Otx is open & OtP is open & x in Otx & P c= OtP & Otx misses OtP by A1,A3,A6,COMPTS_1:15; rng s1 c= OtP by A5,A7,XBOOLE_1:1; then A8: (rng s1) misses Otx by A7,XBOOLE_1:63; A9: R^1 = TopStruct (#the carrier of RealSpace, Family_open_set(RealSpace)#) by PCOMPS_1:def 6,TOPMETR:def 7; then reconsider x as Point of RealSpace; consider r being real number such that A10:r>0 & Ball(x,r) c= Otx by A7,TOPMETR:22,def 7; reconsider r as Real by XREAL_0:def 1; A11: Ball(x,r) misses (rng s1) by A8,A10,XBOOLE_1:63; A12:Ball(x,r) = {q where q is Element of RealSpace :dist(x,q)<r} by METRIC_1:18; not ex n being Nat st for m being Nat st n<=m holds abs(s1.m-(lim s1)) < r proof given n being Nat such that A13:for m being Nat st n<=m holds abs(s1.m-(lim s1)) < r; set m = n + 1; n <= m by NAT_1:29; then abs(s1.m-(lim s1)) < r by A13; then real_dist.(s1.m,lim s1) < r by METRIC_1:def 13; then A14:real_dist.(lim s1,s1.m) < r by METRIC_1:10; reconsider y = s1.m as Element of RealSpace by A9 ,TOPMETR:24; dist(x,y) = (the distance of RealSpace).(x,y) by METRIC_1:def 1; then A15:y in Ball(x,r) by A12,A14,METRIC_1:def 14; s1.m in rng s1 by FUNCT_2:6; then y in Ball(x,r) /\ (rng s1) by A15,XBOOLE_0:def 3; hence thesis by A11,XBOOLE_0:def 7; end; hence thesis by A4,A10,SEQ_2:def 7; end; end; hence thesis by RCOMP_1:def 4; case A16:[#](P) = {}; for s1 being Real_Sequence st (rng s1) c= [#](P) & s1 is convergent holds lim s1 in [#](P) proof let s1 be Real_Sequence; assume A17:(rng s1) c= [#](P) & s1 is convergent; consider x being set such that A18:x in NAT by XBOOLE_0:def 1; s1.x in rng s1 by A18,FUNCT_2:6; hence thesis by A16,A17; end; hence thesis by RCOMP_1:def 4; end; hence thesis; end; theorem Th19: for P being Subset of R^1 holds P is compact implies [#](P) is compact proof let P be Subset of R^1; assume A1:P is compact; now per cases; case [#](P) <> {}; A2:[#](P) is bounded by A1,Th17; [#](P) is closed by A1,Th18; hence thesis by A2,RCOMP_1:29; case A3:[#](P) = {}; assume not [#](P) is compact; then consider s1 being Real_Sequence such that A4:(rng s1) c= [#](P) & not (ex s2 being Real_Sequence st s2 is_subsequence_of s1 & s2 is convergent & (lim s2) in [#](P)) by RCOMP_1:def 3; consider x being set such that A5:x in NAT by XBOOLE_0:def 1; s1.x in rng s1 by A5,FUNCT_2:6; hence thesis by A3,A4; end; hence thesis; end; Lm1: for T being non empty TopSpace holds for f being map of T,R^1 holds for P being Subset of T holds (P <> {} & P is compact & f is continuous) implies ex x1,x2 being Point of T st (x1 in P & x2 in P & f.x1 = upper_bound [#](f.:P) & f.x2 = lower_bound [#](f.:P)) proof let T be non empty TopSpace; let f be map of T,R^1; let P be Subset of T; assume A1:P <> {} & P is compact & f is continuous; A2:[#](f.:P) <> {} proof A3: dom f = the carrier of T by FUNCT_2:def 1; consider x being set such that A4:x in P by A1,XBOOLE_0:def 1; f.x in f.:P by A3,A4,FUNCT_1:def 12; hence thesis by Def3; end; A5:f.:P is compact by A1,Th14; consider y1,y2 being Real such that A6:y1 = upper_bound [#](f.:P) and A7:y2 = lower_bound [#](f.:P); [#](f.:P) is compact by A5,Th19; then A8:y1 in [#](f.:P) & y2 in [#](f.:P) by A2,A6,A7,RCOMP_1:32; A9: [#](f.:P) = f.:P by Def3; then consider x1 being set such that A10:x1 in dom f & x1 in P & y1 = f.x1 by A8,FUNCT_1:def 12; consider x2 being set such that A11:x2 in dom f & x2 in P & y2 = f.x2 by A8,A9,FUNCT_1:def 12; reconsider x1,x2 as Point of T by A10,A11; take x1; take x2; thus thesis by A6,A7,A10,A11; end; definition let P be Subset of R^1; func upper_bound(P) -> Real equals :Def4: upper_bound([#](P)); correctness; func lower_bound(P) -> Real equals :Def5: lower_bound([#](P)); correctness; end; theorem Th20: for T being non empty TopSpace holds for f being map of T,R^1 holds for P being Subset of T holds (P <> {} & P is compact & f is continuous) implies ex x1 being Point of T st (x1 in P & f.x1 = upper_bound(f.:P)) proof let T be non empty TopSpace; let f be map of T,R^1; let P be Subset of T; assume P <> {} & P is compact & f is continuous; then consider x1,x2 being Point of T such that A1:x1 in P & x2 in P & f.x1 = upper_bound [#](f.:P) & f.x2 = lower_bound [#](f.:P) by Lm1; take x1; thus thesis by A1,Def4; end; theorem Th21: for T being non empty TopSpace holds for f being map of T,R^1 holds for P being Subset of T holds (P <> {} & P is compact & f is continuous) implies ex x2 being Point of T st (x2 in P & f.x2 = lower_bound(f.:P)) proof let T be non empty TopSpace; let f be map of T,R^1; let P be Subset of T; assume P <> {} & P is compact & f is continuous; then consider x1,x2 being Point of T such that A1:x1 in P & x2 in P & f.x1 = upper_bound [#](f.:P) & f.x2 = lower_bound [#](f.:P) by Lm1; take x2; thus thesis by A1,Def5; end; begin :: :: The measure of the distance between compact sets :: definition let M be non empty MetrSpace; let x be Point of M; func dist(x) -> map of TopSpaceMetr(M),R^1 means :Def6:for y being Point of M holds it.y = dist(y,x); existence proof defpred EF[Element of M,Element of R^1] means for s being Element of M holds for t being Element of R^1 holds ($1 = s & $2 = t implies t = dist(s,x)); A1:for s being Element of M ex t being Element of R^1 st EF[s,t] proof let s be Element of M; consider t being Real such that A2:t = dist(s,x); reconsider t as Element of R^1 by TOPMETR:24; take t; thus thesis by A2; end; consider F being Function of the carrier of M,the carrier of R^1 such that A3:for x being Element of M holds EF[x,F.x] from FuncExD(A1); A4:for y being Point of M holds F.y = dist(y,x) by A3; TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider F as map of TopSpaceMetr(M),R^1; take F; thus thesis by A4; end; uniqueness proof let F1,F2 be map of TopSpaceMetr(M),R^1; assume A5:for y being Point of M holds F1.y = dist(y,x); assume A6:for y being Point of M holds F2.y = dist(y,x); F1 = F2 proof for y being set st y in the carrier of TopSpaceMetr(M) holds F1.y = F2.y proof let y be set; assume A7:y in the carrier of TopSpaceMetr(M); TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider y as Point of M by A7; F1.y = dist(y,x) by A5 .= F2.y by A6; hence thesis; end; hence thesis by FUNCT_2:18; end; hence thesis; end; end; theorem Th22: for M being non empty MetrSpace holds for x being Point of M holds dist(x) is continuous proof let M be non empty MetrSpace; let x be Point of M; for P being Subset of R^1 st P is open holds (dist(x))"P is open proof let P be Subset of R^1; assume A1:P is open; thus (dist(x))"P is open proof for p being Point of M st p in (dist(x))"P ex r being real number st r>0 & Ball(p,r) c= (dist(x))"P proof let p be Point of M; assume p in (dist(x))"P; then A2: p in dom dist(x) & (dist(x)).p in P by FUNCT_1:def 13; consider y being Point of RealSpace such that A3:y = dist(p,x) by METRIC_1:def 14; reconsider P as Subset of TopSpaceMetr(RealSpace) by TOPMETR:def 7; y in P by A2,A3,Def6; then consider r being real number such that A4:r>0 & Ball(y,r) c= P by A1,TOPMETR:22,def 7; reconsider r as Real by XREAL_0:def 1; A5:Ball(p,r) c= (dist(x))"P proof let z be set; assume A6:z in Ball(p,r); then reconsider z as Point of M; A7:dist(p,z) < r by A6,METRIC_1:12; abs(dist(p,x)-dist(z,x)) <= dist(p,z) by METRIC_6:1; then abs(dist(p,x)-dist(z,x))+dist(p,z) < r+dist(p,z) by A7,REAL_1:67; then A8:abs(dist(p,x)-dist(z,x)) < r by AXIOMS:24; consider q being Point of RealSpace such that A9:q = dist(z,x) by METRIC_1:def 14; dist(y,q) < r by A3,A8,A9,TOPMETR:15; then q in Ball(y,r) by METRIC_1:12; then q in P by A4; then A10:(dist(x)).z in P by A9,Def6; dom (dist(x)) = the carrier of TopSpaceMetr(M) by FUNCT_2:def 1; then dom (dist(x)) = the carrier of M by TOPMETR:16; hence thesis by A10,FUNCT_1:def 13; end; take r; thus thesis by A4,A5; end; hence thesis by TOPMETR:22; end; end; hence thesis by TOPS_2:55; end; theorem Th23: for M being non empty MetrSpace holds for x being Point of M holds for P being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact) implies ex x1 being Point of TopSpaceMetr(M) st (x1 in P & (dist(x)).x1 = upper_bound((dist(x)).:P)) proof let M be non empty MetrSpace; let x be Point of M; let P be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact; dist(x) is continuous by Th22; hence thesis by A1,Th20; end; theorem Th24: for M being non empty MetrSpace holds for x being Point of M holds for P being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact) implies ex x2 being Point of TopSpaceMetr(M) st (x2 in P & (dist(x)).x2 = lower_bound((dist(x)).:P)) proof let M be non empty MetrSpace; let x be Point of M; let P be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact; dist(x) is continuous by Th22; hence thesis by A1,Th21; end; definition let M be non empty MetrSpace; let P be Subset of TopSpaceMetr(M); func dist_max(P) -> map of TopSpaceMetr(M),R^1 means :Def7:for x being Point of M holds it.x = upper_bound((dist(x)).:P); existence proof defpred EF[Element of M,Element of R^1] means for s being Element of M holds for t being Element of R^1 holds ($1 = s & $2 = t implies t = upper_bound((dist(s)).:P)); A1:for s being Element of M ex t being Element of R^1 st EF[s,t] proof let s be Element of M; consider t being Real such that A2:t = upper_bound((dist(s)).:P); reconsider t as Element of R^1 by TOPMETR:24; take t; thus thesis by A2; end; consider F being Function of the carrier of M,the carrier of R^1 such that A3:for x being Element of M holds EF[x,F.x] from FuncExD(A1); A4:for y being Point of M holds F.y = upper_bound((dist(y)).:P) by A3; TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider F as map of TopSpaceMetr(M),R^1; take F; thus thesis by A4; end; uniqueness proof let F1,F2 be map of TopSpaceMetr(M),R^1; assume A5:for y being Point of M holds F1.y = upper_bound((dist(y)).:P); assume A6:for y being Point of M holds F2.y = upper_bound((dist(y)).:P); F1 = F2 proof for y being set st y in the carrier of TopSpaceMetr(M) holds F1.y = F2.y proof let y be set; assume A7:y in the carrier of TopSpaceMetr(M); TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider y as Point of M by A7; F1.y = upper_bound((dist(y)).:P) by A5 .= F2.y by A6; hence thesis; end; hence thesis by FUNCT_2:18; end; hence thesis; end; func dist_min(P) -> map of TopSpaceMetr(M),R^1 means :Def8:for x being Point of M holds it.x = lower_bound((dist(x)).:P); existence proof defpred EF[Element of M,Element of R^1] means for s being Element of M holds for t being Element of R^1 holds ($1 = s & $2 = t implies t = lower_bound((dist(s)).:P)); A8:for s being Element of M ex t being Element of R^1 st EF[s,t] proof let s be Element of M; consider t being Real such that A9:t = lower_bound((dist(s)).:P); reconsider t as Element of R^1 by TOPMETR:24; take t; thus thesis by A9; end; consider F being Function of the carrier of M,the carrier of R^1 such that A10:for x being Element of M holds EF[x,F.x] from FuncExD(A8); A11:for y being Point of M holds F.y = lower_bound((dist(y)).:P) by A10; TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider F as map of TopSpaceMetr(M),R^1; take F; thus thesis by A11; end; uniqueness proof let F1,F2 be map of TopSpaceMetr(M),R^1; assume A12:for y being Point of M holds F1.y = lower_bound((dist(y)).:P); assume A13:for y being Point of M holds F2.y = lower_bound((dist(y)).:P); F1 = F2 proof for y being set st y in the carrier of TopSpaceMetr(M) holds F1.y = F2.y proof let y be set; assume A14:y in the carrier of TopSpaceMetr(M); TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider y as Point of M by A14; F1.y = lower_bound((dist(y)).:P) by A12 .= F2.y by A13; hence thesis; end; hence thesis by FUNCT_2:18; end; hence thesis; end; end; theorem Th25: for M being non empty MetrSpace holds for P being Subset of TopSpaceMetr(M) st P is compact for p1,p2 being Point of M holds p1 in P implies dist(p1,p2) <= upper_bound((dist(p2)).:P) & lower_bound((dist(p2)).:P) <= dist(p1,p2) proof let M be non empty MetrSpace; let P be Subset of TopSpaceMetr(M); assume A1: P is compact; let p1,p2 be Point of M; assume A2:p1 in P; dist(p2) is continuous by Th22; then (dist(p2)).:P is compact by A1,Th14; then A3:[#]((dist(p2)).:P) is bounded by Th17; A4:dist(p1,p2) in [#]((dist(p2)).:P) proof A5:dist(p1,p2) = (dist(p2)).p1 by Def6; dom (dist(p2)) = the carrier of TopSpaceMetr(M) by FUNCT_2:def 1; then (dist(p2)).p1 in (dist(p2)).:P by A2,FUNCT_1:def 12; hence thesis by A5,Def3; end; A6:dist(p1,p2) <= upper_bound((dist(p2)).:P) proof A7:[#]((dist(p2)).:P) is bounded_above by A3,SEQ_4:def 3; upper_bound((dist(p2)).:P) = upper_bound([#]((dist(p2)).:P)) by Def4; hence thesis by A4,A7,SEQ_4:def 4; end; lower_bound((dist(p2)).:P) <= dist(p1,p2) proof A8:[#]((dist(p2)).:P) is bounded_below by A3,SEQ_4:def 3; lower_bound((dist(p2)).:P) = lower_bound([#]((dist(p2)).:P)) by Def5; hence thesis by A4,A8,SEQ_4:def 5; end; hence thesis by A6; end; theorem Th26: for M being non empty MetrSpace holds for P being Subset of TopSpaceMetr(M) st (P <> {} & P is compact) holds for p1,p2 being Point of M holds abs(upper_bound((dist(p1)).:P) - upper_bound((dist(p2)).:P)) <= dist(p1,p2) proof let M be non empty MetrSpace; let P be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact; let p1,p2 be Point of M; consider x1 being Point of TopSpaceMetr(M) such that A2:x1 in P & (dist(p1)).x1 = upper_bound((dist(p1)).:P) by A1,Th23; consider x2 being Point of TopSpaceMetr(M) such that A3:x2 in P & (dist(p2)).x2 = upper_bound((dist(p2)).:P) by A1,Th23; A4: TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider x1 as Point of M; reconsider x2 as Point of M by A4; A5: (dist(p1)).x1 = dist(x1,p1) by Def6; A6:dist(x1,p1) = upper_bound((dist(p1)).:P) by A2,Def6; A7: (dist(p2)).x2 = dist(x2,p2) by Def6; A8:dist(x2,p2) = upper_bound((dist(p2)).:P) by A3,Def6; A9: dist(x1,p1) <= dist(x1,p2) + dist(p2,p1) & dist(x2,p2) <= dist(x2,p1) + dist(p1,p2) by METRIC_1:4; A10:dist(x1,p2) <= dist(x2,p2) by A1,A2,A3,A7,Th25; A11:dist(x2,p1) <= dist(x1,p1) by A1,A2,A3,A5,Th25; dist(x1,p1) - dist(x2,p2) <= dist(x1,p2) + dist(p1,p2) - dist(x1,p2) by A9,A10,REAL_1:92; then A12:dist(x1,p1) - dist(x2,p2) <= dist(p1,p2) by XCMPLX_1:26; dist(x2,p2) - dist(x1,p1) <= dist(x2,p1) + dist(p1,p2) - dist(x2,p1) by A9,A11,REAL_1:92; then dist(x2,p2) - dist(x1,p1) <= dist(p1,p2) by XCMPLX_1:26; then -(dist(x1,p1) - dist(x2,p2)) <= dist(p1,p2) by XCMPLX_1:143; then -dist(p1,p2) <= -(-(dist(x1,p1) - dist(x2,p2))) by REAL_1:50; hence thesis by A6,A8,A12,ABSVALUE:12; end; theorem for M being non empty MetrSpace holds for P being Subset of TopSpaceMetr(M) st (P <> {} & P is compact) holds for p1,p2 being Point of M holds for x1,x2 being Real holds x1 = (dist_max(P)).p1 & x2 = (dist_max(P)).p2 implies abs(x1 - x2) <= dist(p1,p2) proof let M be non empty MetrSpace; let P be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact; let p1,p2 be Point of M; let x1,x2 be Real; assume A2:x1 = (dist_max(P)).p1 & x2 = (dist_max(P)).p2; (dist_max(P)).p1 = upper_bound((dist(p1)).:P) & (dist_max(P)).p2 = upper_bound((dist(p2)).:P) by Def7; hence thesis by A1,A2,Th26; end; theorem Th28: for M being non empty MetrSpace holds for P being Subset of TopSpaceMetr(M) st (P <> {} & P is compact) holds for p1,p2 being Point of M holds abs(lower_bound((dist(p1)).:P) - lower_bound((dist(p2)).:P)) <= dist(p1,p2) proof let M be non empty MetrSpace; let P be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact; let p1,p2 be Point of M; consider x1 being Point of TopSpaceMetr(M) such that A2:x1 in P & (dist(p1)).x1 = lower_bound((dist(p1)).:P) by A1,Th24; consider x2 being Point of TopSpaceMetr(M) such that A3:x2 in P & (dist(p2)).x2 = lower_bound((dist(p2)).:P) by A1,Th24; A4: TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider x1 as Point of M; reconsider x2 as Point of M by A4; A5: (dist(p1)).x1 = dist(x1,p1) by Def6; A6:dist(x1,p1) = lower_bound((dist(p1)).:P) by A2,Def6; A7: (dist(p2)).x2 = dist(x2,p2) by Def6; A8:dist(x2,p2) = lower_bound((dist(p2)).:P) by A3,Def6; A9: dist(x1,p2) <= dist(x1,p1) + dist(p1,p2) & dist(x2,p1) <= dist(x2,p2) + dist(p2,p1) by METRIC_1:4; A10: dist(x2,p2) <= dist(x1,p2) by A1,A2,A3,A7,Th25; dist(x1,p1) <= dist(x2,p1) by A1,A2,A3,A5,Th25; then dist(x1,p1) <= dist(x2,p2) + dist(p1,p2) by A9,AXIOMS:22; then A11:dist(x1,p1) - dist(x2,p2) <= dist(p1,p2) by REAL_1:86; dist(x2,p2) <= dist(x1,p1) + dist(p1,p2) by A9,A10,AXIOMS:22; then dist(x2,p2) - dist(x1,p1) <= dist(p1,p2) by REAL_1:86; then -(dist(x1,p1) - dist(x2,p2)) <= dist(p1,p2) by XCMPLX_1:143; then -dist(p1,p2) <= -(-(dist(x1,p1) - dist(x2,p2))) by REAL_1:50; hence thesis by A6,A8,A11,ABSVALUE:12; end; theorem for M being non empty MetrSpace holds for P being Subset of TopSpaceMetr(M) st (P <> {} & P is compact) holds for p1,p2 being Point of M holds for x1,x2 being Real holds x1 = (dist_min(P)).p1 & x2 = (dist_min(P)).p2 implies abs(x1 - x2) <= dist(p1,p2) proof let M be non empty MetrSpace; let P be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact; let p1,p2 be Point of M; let x1,x2 be Real; assume A2:x1 = (dist_min(P)).p1 & x2 = (dist_min(P)).p2; (dist_min(P)).p1 = lower_bound((dist(p1)).:P) & (dist_min(P)).p2 = lower_bound((dist(p2)).:P) by Def8; hence thesis by A1,A2,Th28; end; theorem Th30: for M being non empty MetrSpace holds for X being Subset of TopSpaceMetr(M) st (X <> {} & X is compact) holds dist_max(X) is continuous proof let M be non empty MetrSpace; let X be Subset of TopSpaceMetr(M); assume A1:X <> {} & X is compact; for P being Subset of R^1 st P is open holds (dist_max(X))"P is open proof let P be Subset of R^1; assume A2:P is open; for p being Point of M st p in (dist_max(X))"P ex r being real number st r>0 & Ball(p,r) c= (dist_max(X))"P proof let p be Point of M; assume p in (dist_max(X))"P; then A3: p in dom dist_max(X) & (dist_max(X)).p in P by FUNCT_1:def 13; consider y being Point of RealSpace such that A4:y = upper_bound((dist(p)).:(X)) by METRIC_1:def 14; reconsider P as Subset of TopSpaceMetr(RealSpace) by TOPMETR:def 7; y in P by A3,A4,Def7; then consider r being real number such that A5:r>0 & Ball(y,r) c= P by A2,TOPMETR:22,def 7; reconsider r as Real by XREAL_0:def 1; A6:Ball(p,r) c= (dist_max(X))"P proof let z be set; assume A7:z in Ball(p,r); then reconsider z as Point of M; A8:dist(p,z) < r by A7,METRIC_1:12; abs(upper_bound((dist(p)).:(X)) - upper_bound((dist(z)).:(X))) <= dist(p,z) by A1,Th26; then abs(upper_bound((dist(p)).:(X)) - upper_bound((dist(z)).:(X)))+dist(p,z) < r+dist(p,z) by A8,REAL_1:67; then A9:abs(upper_bound((dist(p)).:(X)) - upper_bound((dist(z)).:(X))) < r by AXIOMS:24; consider q being Point of RealSpace such that A10:q = upper_bound((dist(z)).:(X)) by METRIC_1:def 14; A11:q = (dist_max(X)).z by A10,Def7; dist(y,q) < r by A4,A9,A10,TOPMETR:15; then A12: q in Ball(y,r) by METRIC_1:12; dom (dist_max(X)) = the carrier of TopSpaceMetr(M) by FUNCT_2:def 1; then dom (dist_max(X)) = the carrier of M by TOPMETR:16; hence thesis by A5,A11,A12,FUNCT_1:def 13; end; take r; thus thesis by A5,A6; end; hence thesis by TOPMETR:22; end; hence thesis by TOPS_2:55; end; theorem Th31: for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact & Q <> {} & Q is compact) implies ex x1 being Point of TopSpaceMetr(M) st (x1 in Q & (dist_max(P)).x1 = upper_bound((dist_max(P)).:Q)) proof let M be non empty MetrSpace; let P,Q be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact & Q <> {} & Q is compact; then dist_max(P) is continuous by Th30; then consider x1 being Point of TopSpaceMetr(M) such that A2:x1 in Q & (dist_max(P)).x1 = upper_bound((dist_max(P)).:Q) by A1,Th20; take x1; thus thesis by A2; end; theorem Th32: for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact & Q <> {} & Q is compact) implies ex x2 being Point of TopSpaceMetr(M) st (x2 in Q & (dist_max(P)).x2 = lower_bound((dist_max(P)).:Q)) proof let M be non empty MetrSpace; let P,Q be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact & Q <> {} & Q is compact; then dist_max(P) is continuous by Th30; then consider x2 being Point of TopSpaceMetr(M) such that A2:x2 in Q & (dist_max(P)).x2 = lower_bound((dist_max(P)).:Q) by A1,Th21; take x2; thus thesis by A2; end; theorem Th33: for M being non empty MetrSpace holds for X being Subset of TopSpaceMetr(M) st X <> {} & X is compact holds dist_min(X) is continuous proof let M be non empty MetrSpace; let X be Subset of TopSpaceMetr(M); assume A1:X <> {} & X is compact; for P being Subset of R^1 st P is open holds (dist_min(X))"P is open proof let P be Subset of R^1; assume A2:P is open; thus (dist_min(X))"P is open proof for p being Point of M st p in (dist_min(X))"P ex r being real number st r>0 & Ball(p,r) c= (dist_min(X))"P proof let p be Point of M; assume A3:p in (dist_min(X))"P; ex r being Real st r>0 & Ball(p,r) c= (dist_min(X))"P proof A4: p in dom dist_min(X) & (dist_min(X)).p in P by A3,FUNCT_1:def 13; consider y being Point of RealSpace such that A5:y = lower_bound((dist(p)).:(X)) by METRIC_1:def 14; reconsider P as Subset of TopSpaceMetr(RealSpace) by TOPMETR:def 7; y in P by A4,A5,Def8; then consider r being real number such that A6:r>0 & Ball(y,r) c= P by A2,TOPMETR:22,def 7; reconsider r as Real by XREAL_0:def 1; A7:Ball(p,r) c= (dist_min(X))"P proof let z be set; assume A8:z in Ball(p,r); then reconsider z as Point of M; A9:dist(p,z) < r by A8,METRIC_1:12; abs(lower_bound((dist(p)).:(X)) - lower_bound((dist(z)).:(X))) <= dist(p,z) by A1,Th28; then abs(lower_bound((dist(p)).:(X)) - lower_bound((dist(z)).:(X)))+dist(p,z) < r+dist(p,z) by A9,REAL_1:67; then A10:abs(lower_bound((dist(p)).:(X)) - lower_bound((dist(z)).:(X))) < r by AXIOMS:24; consider q being Point of RealSpace such that A11:q = lower_bound((dist(z)).:(X)) by METRIC_1:def 14; A12:q = (dist_min(X)).z by A11,Def8; dist(y,q) < r by A5,A10,A11,TOPMETR:15; then A13: q in Ball(y,r) by METRIC_1:12; dom (dist_min(X)) = the carrier of TopSpaceMetr(M) by FUNCT_2:def 1; then dom (dist_min(X)) = the carrier of M by TOPMETR:16; hence thesis by A6,A12,A13,FUNCT_1:def 13; end; take r; thus thesis by A6,A7; end; hence thesis; end; hence thesis by TOPMETR:22; end; end; hence thesis by TOPS_2:55; end; theorem Th34: for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact & Q <> {} & Q is compact) implies ex x1 being Point of TopSpaceMetr(M) st (x1 in Q & (dist_min(P)).x1 = upper_bound((dist_min(P)).:Q)) proof let M be non empty MetrSpace; let P,Q be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact & Q <> {} & Q is compact; then dist_min(P) is continuous by Th33; then consider x1 being Point of TopSpaceMetr(M) such that A2:x1 in Q & (dist_min(P)).x1 = upper_bound((dist_min(P)).:Q) by A1,Th20; take x1; thus thesis by A2; end; theorem Th35: for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact & Q <> {} & Q is compact) implies ex x2 being Point of TopSpaceMetr(M) st (x2 in Q & (dist_min(P)).x2 = lower_bound((dist_min(P)).:Q)) proof let M be non empty MetrSpace; let P,Q be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact & Q <> {} & Q is compact; then dist_min(P) is continuous by Th33; then consider x2 being Point of TopSpaceMetr(M) such that A2:x2 in Q &(dist_min(P)).x2 = lower_bound((dist_min(P)).:Q) by A1,Th21; take x2; thus thesis by A2; end; definition let M be non empty MetrSpace; let P,Q be Subset of TopSpaceMetr(M); func min_dist_min(P,Q) -> Real equals :Def9: lower_bound((dist_min(P)).:Q); correctness; func max_dist_min(P,Q) -> Real equals :Def10: upper_bound((dist_min(P)).:Q); correctness; func min_dist_max(P,Q) -> Real equals :Def11: lower_bound((dist_max(P)).:Q); correctness; func max_dist_max(P,Q) -> Real equals :Def12: upper_bound((dist_max(P)).:Q); correctness; end; theorem for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is compact & Q <> {} & Q is compact holds ex x1,x2 being Point of M st x1 in P & x2 in Q & dist(x1,x2) = min_dist_min(P,Q) proof let M be non empty MetrSpace; let P,Q be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact & Q <> {} & Q is compact; then consider x2 being Point of TopSpaceMetr(M) such that A2:x2 in Q & (dist_min(P)).x2 = lower_bound((dist_min(P)).:Q) by Th35; A3: TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider x2 as Point of M; consider x1 being Point of TopSpaceMetr(M) such that A4:x1 in P & (dist(x2)).x1 = lower_bound((dist(x2)).:P) by A1,Th24; reconsider x1 as Point of M by A3; take x1; take x2; dist(x1,x2) = (dist(x2)).x1 by Def6 .= lower_bound((dist_min(P)).:Q) by A2,A4,Def8; hence thesis by A2,A4,Def9; end; theorem for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is compact & Q <> {} & Q is compact holds ex x1,x2 being Point of M st x1 in P & x2 in Q & dist(x1,x2) = min_dist_max(P,Q) proof let M be non empty MetrSpace; let P,Q be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact & Q <> {} & Q is compact; then consider x2 being Point of TopSpaceMetr(M) such that A2:x2 in Q & (dist_max(P)).x2 = lower_bound((dist_max(P)).:Q) by Th32; A3: TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider x2 as Point of M; consider x1 being Point of TopSpaceMetr(M) such that A4:x1 in P & (dist(x2)).x1 = upper_bound((dist(x2)).:P) by A1,Th23; reconsider x1 as Point of M by A3; take x1; take x2; dist(x1,x2) = (dist(x2)).x1 by Def6 .= lower_bound((dist_max(P)).:Q) by A2,A4,Def7; hence thesis by A2,A4,Def11; end; theorem for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is compact & Q <> {} & Q is compact holds ex x1,x2 being Point of M st x1 in P & x2 in Q & dist(x1,x2) = max_dist_min(P,Q) proof let M be non empty MetrSpace; let P,Q be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact & Q <> {} & Q is compact; then consider x2 being Point of TopSpaceMetr(M) such that A2:x2 in Q & (dist_min(P)).x2 = upper_bound((dist_min(P)).:Q) by Th34; A3: TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider x2 as Point of M; consider x1 being Point of TopSpaceMetr(M) such that A4:x1 in P & (dist(x2)).x1 = lower_bound((dist(x2)).:P) by A1,Th24; reconsider x1 as Point of M by A3; take x1; take x2; dist(x1,x2) = (dist(x2)).x1 by Def6 .= upper_bound((dist_min(P)).:Q) by A2,A4,Def8; hence thesis by A2,A4,Def10; end; theorem for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is compact & Q <> {} & Q is compact holds ex x1,x2 being Point of M st x1 in P & x2 in Q & dist(x1,x2) = max_dist_max(P,Q) proof let M be non empty MetrSpace; let P,Q be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact & Q <> {} & Q is compact; then consider x2 being Point of TopSpaceMetr(M) such that A2:x2 in Q & (dist_max(P)).x2 = upper_bound((dist_max(P)).:Q) by Th31; A3: TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) by PCOMPS_1:def 6; then reconsider x2 as Point of M; consider x1 being Point of TopSpaceMetr(M) such that A4:x1 in P & (dist(x2)).x1 = upper_bound((dist(x2)).:P) by A1,Th23; reconsider x1 as Point of M by A3; take x1; take x2; dist(x1,x2) = (dist(x2)).x1 by Def6 .= upper_bound((dist_max(P)).:Q) by A2,A4,Def7; hence thesis by A2,A4,Def12; end; theorem for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) st P is compact & Q is compact holds for x1,x2 being Point of M st x1 in P & x2 in Q holds min_dist_min(P,Q) <= dist(x1,x2) & dist(x1,x2) <= max_dist_max(P,Q) proof let M be non empty MetrSpace; let P,Q be Subset of TopSpaceMetr(M); assume A1: P is compact & Q is compact; let x1,x2 be Point of M; assume A2:x1 in P & x2 in Q; A3: dom (dist_max(P)) = the carrier of TopSpaceMetr(M) by FUNCT_2:def 1; x2 in the carrier of M; then x2 in the carrier of TopSpaceMetr(M) by TOPMETR:16; then (dist_max(P)).x2 in the carrier of R^1 by FUNCT_2:7; then consider y being Real such that A4:y = (dist_max(P)).x2 by TOPMETR:24; A5:dist(x1,x2) <= upper_bound((dist(x2)).:P) & lower_bound((dist(x1)).:Q) <= dist(x2,x1) by A1,A2,Th25; A6: upper_bound((dist(x2)).:P) = y by A4,Def7; dist_max(P) is continuous by A1,A2,Th30; then (dist_max(P)).:Q is compact by A1,Th14; then A7:[#]((dist_max(P)).:Q) is bounded by Th17; A8:y in [#]((dist_max(P)).:Q) proof (dist_max(P)).x2 in (dist_max(P)).:Q by A2,A3,FUNCT_1:def 12; hence thesis by A4,Def3; end; y <= upper_bound((dist_max(P)).:Q) proof A9:[#]((dist_max(P)).:Q) is bounded_above by A7,SEQ_4:def 3; upper_bound((dist_max(P)).:Q) = upper_bound([#]((dist_max(P)).:Q)) by Def4; hence thesis by A8,A9,SEQ_4:def 4; end; then A10: dist(x1,x2) <= upper_bound((dist_max(P)).:Q) by A5,A6,AXIOMS:22; A11: dom (dist_min(P)) = the carrier of TopSpaceMetr(M) by FUNCT_2:def 1; x2 in the carrier of M; then x2 in the carrier of TopSpaceMetr(M) by TOPMETR:16; then (dist_min(P)).x2 in the carrier of R^1 by FUNCT_2:7; then consider z being Real such that A12:z = (dist_min(P)).x2 by TOPMETR:24; A13:dist(x1,x2) <= upper_bound((dist(x2)).:P) & lower_bound((dist(x2)).:P) <= dist(x1,x2) by A1,A2,Th25; A14: lower_bound((dist(x2)).:P) = z by A12,Def8; dist_min(P) is continuous by A1,A2,Th33; then (dist_min(P)).:Q is compact by A1,Th14; then A15:[#]((dist_min(P)).:Q) is bounded by Th17; A16:z in [#]((dist_min(P)).:Q) proof (dist_min(P)).x2 in (dist_min(P)).:Q by A2,A11,FUNCT_1:def 12; hence thesis by A12,Def3; end; lower_bound((dist_min(P)).:Q) <= z proof A17:[#]((dist_min(P)).:Q) is bounded_below by A15,SEQ_4:def 3; lower_bound((dist_min(P)).:Q) = lower_bound([#]((dist_min(P)).:Q)) by Def5; hence thesis by A16,A17,SEQ_4:def 5; end; then lower_bound((dist_min(P)).:Q) <= dist(x1,x2) by A13,A14,AXIOMS:22; hence thesis by A10,Def9,Def12; end; :: B i b l i o g r a p h y :: J.Dieudonne, "Foundations of Modern Analysis",Academic Press, 1960. :: J.L.Kelley, "General Topology", von Nostnend, 1955. ::

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