### The Mizar article:

### Basic Properties of Rational Numbers

**by****Andrzej Kondracki**

- Received July 10, 1990
Copyright (c) 1990 Association of Mizar Users

- MML identifier: RAT_1
- [ MML identifier index ]

environ vocabulary ARYTM, INT_1, ABSVALUE, ARYTM_3, ARYTM_1, RELAT_1, RAT_1, ORDINAL2, BOOLE, COMPLEX1, OPPCAT_1, ARYTM_2, QUOFIELD, ORDINAL1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, ORDINAL2, ARYTM_3, ARYTM_2, NUMBERS, ARYTM_0, XCMPLX_0, XREAL_0, REAL_1, NAT_1, ABSVALUE, INT_1, ARYTM_1; constructors NAT_1, XCMPLX_0, XREAL_0, ORDINAL2, INT_1, ABSVALUE, XBOOLE_0, ARYTM_3, ARYTM_0, FUNCT_4, ARYTM_2, ARYTM_1, REAL_1; clusters INT_1, XREAL_0, REAL_1, ZFMISC_1, XBOOLE_0, NUMBERS, ARYTM_3, ORDINAL2, ARYTM_2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; definitions XREAL_0, TARSKI, XBOOLE_0; theorems AXIOMS, REAL_1, INT_1, NAT_1, ABSVALUE, TARSKI, XREAL_0, XCMPLX_0, XCMPLX_1, ORDINAL2, XBOOLE_0, NUMBERS, ARYTM_3, ARYTM_0, ARYTM_2, ZFMISC_1, ARYTM_1; schemes NAT_1; begin Lm0: one = succ 0 by ORDINAL2:def 4 .= 1; reserve X,x for set, a,b,c,d for real number, k,k1,l,l1 for Nat, m,m1,n,n1 for Integer; :: Auxiliary theorems & redefinitions. definition let i be integer number; cluster abs i -> natural; coherence proof 0 <= abs i by ABSVALUE:5; hence thesis by INT_1:16; end; end; definition let k; redefine func abs k -> Nat; coherence by ORDINAL2:def 21; end; Lm1:0<b & 0<d implies (a*d<c*b iff a/b<c/d) proof assume A1:0<b & 0<d; then A2:0<b" & 0<d" by REAL_1:72; A3:now assume a*d<c*b; then (a*d)*d"<(c*b)*d" by A2,REAL_1:70; then a*(d*d")<(c*b)*d" by XCMPLX_1:4; then a*1<(c*b)*d" by A1,XCMPLX_0:def 7; then a<b*(c*d") by XCMPLX_1:4; then a<b*(c/d) by XCMPLX_0:def 9; then b"*a<b"*(b*(c/d)) by A2,REAL_1:70; then b"*a<(b"*b)*(c/d) by XCMPLX_1:4; then b"*a<1*(c/d) by A1,XCMPLX_0:def 7; hence a/b<c/d by XCMPLX_0:def 9; end; now assume a/b<c/d; then a/b*b<c/d*b by A1,REAL_1:70; then a<b*(c/d) by A1,XCMPLX_1:88; then a*d<(b*(c/d))*d by A1,REAL_1:70; then a*d<b*(c/d*d) by XCMPLX_1:4; hence a*d<c*b by A1,XCMPLX_1:88; end; hence thesis by A3; end; :: We define RAT as a set of real numbers, each of which can be expressed :: as a quotient of integer dividend and divisor (divisor is not equal :: to zero). reserve i,j,i1,j1 for Nat; Lm_2: quotient(i1,j1) = i1/j1 proof set x = quotient(i1,j1); per cases; suppose S: j1 = 0; hence x = 0 by ARYTM_3:def 9 .= i1/j1 by S,XCMPLX_1:49; suppose S2: j1 <> 0; j1 * j1" = 1 by XCMPLX_0:def 7,S2; then consider x1,x2,y1,y2 being Real such that W1: j1 = [*x1,x2*] and W2: j1" = [*y1,y2*] and W3: 1 = [* +(*(x1,y1),opp*(x2,y2)), +(*(x1,y2),*(x2,y1)) *] by XCMPLX_0:def 5; X2: x2 = 0 by W1,ARYTM_0:26; then X1: j1 = x1 by W1,ARYTM_0:def 7; y2 = 0 by W2,ARYTM_0:26; then Y1: j1" = y1 by W2,ARYTM_0:def 7; +(0,opp 0) = 0 by ARYTM_0:def 4; then OP: opp 0 = 0 by ARYTM_0:13; +(*(x1,y2),*(x2,y1)) = 0 by W3,ARYTM_0:26; then XY1: 1 = +(*(x1,y1),opp*(x2,y2)) by W3,ARYTM_0:def 7 .= +(*(x1,y1),opp 0) by ARYTM_0:14,X2 .= *(x1,y1) by ARYTM_0:13,OP; consider x'1,x'2,y'1,y'2 being Real such that W4: i1 = [*x'1,x'2*] and W5: j1" = [*y'1,y'2*] and W6: i1*j1" = [* +(*(x'1,y'1),opp*(x'2,y'2)), +(*(x'1,y'2),*(x'2,y'1)) *] by XCMPLX_0:def 5; x'2 = 0 by W4,ARYTM_0:26; then X'1: i1 = x'1 by W4,ARYTM_0:def 7; Y'2: y'2 = 0 by W5,ARYTM_0:26; then Y'1: j1" = y'1 by W5,ARYTM_0:def 7; +(*(x'1,y'2),*(x'2,y'1)) = 0 by W6,ARYTM_0:26; then XY'1: i1*j1" = +(*(x'1,y1),opp*(x'2,0)) by W6,ARYTM_0:def 7,Y'1,Y1,Y'2 .= +(*(x'1,y1),0) by ARYTM_0:14,OP .= *(x'1,y1) by ARYTM_0:13; x'1 in omega by X'1; then B: x'1 in RAT+ by ARYTM_3:34; x1 in omega by X1; then reconsider xx = x1 as Element of RAT+ by ARYTM_3:34; consider yy being Element of RAT+ such that WW: xx *' yy = one by ARYTM_3:60,S2,X1; xx in RAT+ & yy in RAT+; then reconsider xx1 = xx, yy1 = yy as Element of REAL+ by ARYTM_2:1; I: ex x',y' being Element of RAT+ st xx1 = x' & yy1 = y' & xx1 *' yy1 = x' *' y' by ARYTM_2:22; xx1 in REAL+ & yy1 in REAL+; then reconsider xx1, yy1 as Element of REAL by ARYTM_0:1; ex x',y' being Element of REAL+ st xx1 = x' & yy1 = y' & *(xx1,yy1) = x' *' y' by ARYTM_0:def 3; then yy = y1 by S2,XY1,Lm0,ARYTM_0:20,X1,I,WW; then G: y1 in RAT+; then consider x',y' being Element of REAL+ such that W7: x'1 = x' and W8: y1 = y' and W9: i1 * j1" = x' *' y' by B,ARYTM_0:def 3,XY'1,ARYTM_2:1; consider x'',y'' being Element of RAT+ such that W10: x' = x'' and W11: y' = y'' and W12: i1 * j1" = x'' *' y'' by B,ARYTM_2:22,W9,W7,W8,G; HH: j1 in omega; then reconsider a = x, b = j1 as Element of RAT+ by ARYTM_3:34; x1 in omega by X1; then consider x1',y1' being Element of REAL+ such that W13: x1 = x1' and W14: y1 = y1' and W15: *(x1,y1) = x1' *' y1' by ARYTM_0:def 3,G,ARYTM_2:1,2; ex x1'',y1'' being Element of RAT+ st x1' = x1'' & y1' = y1'' & x1' *' y1' = x1'' *' y1'' by ARYTM_2:22,W13,W14,G,X1,ARYTM_3:34,HH; then XX: b *' y'' = 1 by XY1,W15,X1,W14,W13,W11,W8; a *' b = quotient(i1,j1) *' quotient(j1,one) by ARYTM_3:46 .= quotient(i1*^j1,j1*^one) by ARYTM_3:55 .= quotient(i1,one) *' quotient(j1,j1) by ARYTM_3:55 .= quotient(i1,one) *' one by S2, ARYTM_3:47 .= quotient(i1,one) by ARYTM_3:59 .= i1 by ARYTM_3:46 .= x'' *' one by ARYTM_3:59,W10,W7,X'1 .= x'' *' y'' *' b by ARYTM_3:58,XX,Lm0; hence x = i1 * j1" by ARYTM_3:62,S2,W12 .= i1/j1 by XCMPLX_0:def 9; end; 0 in omega; then reconsider 0' = 0 as Element of REAL+ by ARYTM_2:2; Lm_3: for x' being Element of REAL+ st x' = a holds 0' - x' = -a proof let xx being Element of REAL+ such that Z: xx = a; per cases; suppose xx = 0; hence 0' - xx = -a by Z,ARYTM_1:18; suppose S: xx <> 0; set b = 0' - xx; U: 0' <=' xx by ARYTM_1:6; not xx <=' 0' by U,S,ARYTM_1:4; then X: b = [{},xx -' 0'] by ARYTM_1:def 2; now assume b = [0,0]; then xx -' 0' = 0 by X,ZFMISC_1:33; hence contradiction by S,U,ARYTM_1:10; end; then Y: not b in {[0,0]} by TARSKI:def 1; 0 in {0} by TARSKI:def 1; then V: b in [:{0},REAL+:] by X,ZFMISC_1:106; then b in REAL+ \/ [:{0},REAL+:] by XBOOLE_0:def 2; then reconsider b as Element of REAL by Y,XBOOLE_0:def 4,NUMBERS:def 1; consider x1,x2,y1,y2 being Element of REAL such that W1: a = [*x1,x2*] and W2: b = [*y1,y2*] and W3: a+b = [*+(x1,y1),+(x2,y2)*] by XCMPLX_0:def 4; a in REAL by XREAL_0:def 1; then x2 = 0 by W1,ARYTM_0:26; then X1: a = x1 by W1,ARYTM_0:def 7; y2 = 0 by W2,ARYTM_0:26; then Y1: b = y1 by W2,ARYTM_0:def 7; a+b in REAL by XREAL_0:def 1; then +(x2,y2) = 0 by W3,ARYTM_0:26; then XY1: a+b = +(x1,y1) by W3,ARYTM_0:def 7; y1 in [:{0},REAL+:] by Y1,V; then consider x',y' being Element of REAL+ such that W4: x1 = x' and W5: y1 = [0,y'] and W6: +(x1,y1) = x' - y' by ARYTM_0:def 2,Z,X1; P: y' = xx -' 0' by W5,Y1,X,ZFMISC_1:33; Q: xx -' 0' = xx -' 0' + 0' by ARYTM_2:def 8 .= xx by ARYTM_1:def 1,U; a + b = +(x1,y1) by XY1 .= 0 by ARYTM_1:18,W6,P,W4,X1,Z,Q; hence 0' - xx = -a by XCMPLX_0:def 6; end; Lm_0: x in RAT implies ex m,n st x = m/n proof assume Z: x in RAT; pc: x in RAT+ \/ [:{0},RAT+:] by Z,XBOOLE_0:def 4,NUMBERS:def 3; per cases by pc,XBOOLE_0:def 2; suppose x in RAT+; then reconsider x as Element of RAT+; take numerator x,denominator x; quotient(numerator x,denominator x) = x by ARYTM_3:45; hence thesis by Lm_2; suppose x in [:{0},RAT+:]; then consider x1,x2 being set such that W1: x1 in {0} and W2: x2 in RAT+ and W3: x = [x1,x2] by ZFMISC_1:103; reconsider x2 as Element of RAT+ by W2; reconsider x' = x2 as Element of REAL+ by W2,ARYTM_2:1; take m = -numerator x2, n = denominator x2; A: x2 = quotient(numerator x2,denominator x2) by ARYTM_3:45 .= numerator x2/n by Lm_2; x1 = 0 by W1,TARSKI:def 1; then C: x = [0,x'] by W3; not x in {[0,0]} by Z,XBOOLE_0:def 4,NUMBERS:def 3; then x <> [0,0] by TARSKI:def 1; then B: x2 <> 0 by C; thus x = 0' - x' by C,ARYTM_1:19,B .= -numerator x2/n by Lm_3,A .= m/n by XCMPLX_1:188; end; Lm_1': x = m/l implies x in RAT proof assume Z: x = m/l; then x in REAL by XREAL_0:def 1; then A: not x in {[0,0]} by NUMBERS:def 1,XBOOLE_0:def 4; m is Element of INT by INT_1:def 2; then K: ex k st m = k or m = -k by INT_1:def 1; per cases; suppose l = 0; then x = m * 0" by Z,XCMPLX_0:def 9 .= 0; then x in omega; then x in RAT+ by ARYTM_3:34; then x in RAT+ \/ [:{0},RAT+:] by XBOOLE_0:def 2; hence thesis by A,XBOOLE_0:def 4,NUMBERS:def 3; suppose ex k st m = k; then consider k such that W: m = k; x = quotient(k,l) by Z,Lm_2,W; then x in RAT+ \/ [:{0},RAT+:] by XBOOLE_0:def 2; hence x in RAT by XBOOLE_0:def 4,NUMBERS:def 3,A; suppose that S1: l <> 0 and S2: for k holds m <> k; consider k such that W: m = -k by K,S2; B: x = -k/l by W,Z, XCMPLX_1:188; k/l = quotient(k,l) by Lm_2; then E: k/l in RAT+; then k/l in REAL+ by ARYTM_2:1; then reconsider x' = k/l as Element of REAL+; m <> 0 by S2; then C: x' <> 0 by B,Z,XCMPLX_1:50,S1; D: x = 0' - x' by Lm_3,B .= [0,x'] by ARYTM_1:19,C; 0 in {0} by TARSKI:def 1; then x in [:{0},RAT+:] by D,ZFMISC_1:106,E; then x in RAT+ \/ [:{0},RAT+:] by XBOOLE_0:def 2; hence x in RAT by A,XBOOLE_0:def 4,NUMBERS:def 3; end; Lm_1: x = m/n implies x in RAT proof assume Z: x = m/n; n is Element of INT by INT_1:def 2; then consider k such that W: n = k or n = -k by INT_1:def 1; per cases by W; suppose n = k; hence x in RAT by Z,Lm_1'; suppose n = -k; then x = (-m)/k by Z, XCMPLX_1:193; hence x in RAT by Lm_1'; end; definition redefine func RAT means :Def1: x in it iff ex m,n st x = m/n; compatibility proof let R be set; thus R = RAT implies for x holds x in R iff ex m,n st x = m/n by Lm_0,Lm_1; assume Z: for x holds x in R iff ex m,n st x = m/n; thus R c= RAT proof let x; assume x in R; then ex m,n st x = m/n by Z; hence thesis by Lm_1; end; let x; assume x in RAT; then ex m,n st x = m/n by Lm_0; hence thesis by Z; end; end; definition let r be number; attr r is rational means :Def2: r in RAT; end; definition cluster rational Real; existence proof reconsider a = 0, b = 1 as Integer; take x=a/b; thus x is Real; thus x in RAT by Def1; end; end; definition cluster rational number; existence proof reconsider a = 0 as Integer; reconsider b = 1 as Integer; take x=a/b; thus x in RAT by Def1; end; end; definition mode Rational is rational number; end; theorem Th1: x in RAT implies ex m,n st n<>0 & x = m/n proof assume A1: x in RAT; per cases; suppose x = 0; then x = 0/1; hence thesis; suppose A2: x <> 0; consider m,n such that A3: x = m/n by A1,Def1; take m,n; A4: x = m*n" by A3,XCMPLX_0:def 9; hereby assume n = 0; then n" = 0 by XCMPLX_0:def 7; hence contradiction by A2,A4; end; thus thesis by A3; end; canceled; theorem Th3: x is Rational implies ex m,n st n<>0 & x = m/n proof assume x is Rational; then x in RAT by Def2; hence thesis by Th1; end; theorem Th4: RAT c= REAL by NUMBERS:12; definition cluster rational -> real number; coherence proof let n be number; assume n in RAT; hence n in REAL by Th4; end; end; canceled; theorem Th6: (ex m,n st x = m/n) implies x is rational proof x is Rational iff x in RAT by Def2; hence thesis by Def1; end; theorem Th7: for x being Integer holds x is Rational proof let x be Integer; x = x/1; hence thesis by Th6; end; definition cluster integer -> rational number; coherence by Th7; end; canceled 3; theorem INT c= RAT by NUMBERS:14; theorem NAT c= RAT by NUMBERS:18; reserve p,q for Rational; :: Now we prove that fractions of integer denominator and natural numerator :: or of natural denominator and numerator, etc., are all rational numbers. definition let p,q; cluster p*q -> rational; coherence proof consider m1,n1 being Integer such that n1<>0 and A2: p = m1/n1 by Th3; consider m2,n2 being Integer such that n2<>0 and A4: q = m2/n2 by Th3; p*q=(m1*m2)/(n1*n2) by A2,A4,XCMPLX_1:77; hence thesis by Th6; end; cluster p+q -> rational; coherence proof consider m1,n1 being Integer such that A6: n1<>0 and A7: p = m1/n1 by Th3; consider m2,n2 being Integer such that A8: n2<>0 and A9: q = m2/n2 by Th3; p+q=(m1*n2+m2*n1)/(n1*n2) by A6,A7,A8,A9,XCMPLX_1:117; hence thesis by Th6; end; cluster p-q -> rational; coherence proof consider m1,n1 being Integer such that A11: n1<>0 and A12: p = m1/n1 by Th3; consider m2,n2 being Integer such that A13: n2<>0 and A14: q = m2/n2 by Th3; p-q=(m1*n2-m2*n1)/(n1*n2) by A11,A12,A13,A14,XCMPLX_1:131; hence thesis by Th6; end; end; definition let p,m; cluster p+m -> rational; coherence proof reconsider x=m as Rational by Th7; p+x is Rational;hence thesis; end; cluster p-m -> rational; coherence proof reconsider x=m as Rational by Th7; p-x is Rational;hence thesis; end; cluster p*m -> rational; coherence proof reconsider x=m as Rational by Th7; p*x is Rational;hence thesis; end; end; definition let m,p; cluster m+p -> rational; coherence proof reconsider x=m as Rational by Th7; x+p is Rational;hence thesis; end; cluster m-p -> rational; coherence proof reconsider x=m as Rational by Th7; x-p is Rational;hence thesis; end; cluster m*p -> rational; coherence proof reconsider x=m as Rational by Th7; x*p is Rational;hence thesis; end; end; definition let p,k; cluster p+k -> rational; coherence; cluster p-k -> rational; coherence; cluster p*k -> rational; coherence; end; definition let k,p; cluster k+p -> rational; coherence; cluster k-p -> rational; coherence; cluster k*p -> rational; coherence; end; definition let p; cluster -p -> rational; coherence proof consider m,n such that n<>0 and A2: p=m/n by Th3; -p = (-m)/n by A2,XCMPLX_1:188; hence thesis by Th6; end; cluster abs(p) -> rational; coherence proof consider m,n such that A3:n<>0 & p=m/n by Th3; abs(p)=abs(m)/abs(n) by A3,ABSVALUE:16; hence thesis by Th6; end; end; :: Now we prove that the quotient of two rational numbers is rational canceled 3; theorem Th16: for p,q holds p/q is Rational proof let p,q; consider m1,n1 being Integer such that n1<>0 and A3: p = m1/n1 by Th3; consider m2,n2 being Integer such that n2<>0 and A4: q = m2/n2 by Th3; p/q = p*(m2/n2)" by A4,XCMPLX_0:def 9 .= p*(1/(m2/n2)) by XCMPLX_1:217 .= p*((1*n2)/m2) by XCMPLX_1:78 .= (m1*n2)/(n1*m2) by A3,XCMPLX_1:77; hence thesis by Th6; end; definition let p, q be rational number; cluster p/q -> rational; coherence by Th16; end; canceled 4; theorem Th21: p" is Rational proof p" = 1/p by XCMPLX_1:217; hence thesis; end; definition let p be rational number; cluster p" -> rational; coherence by Th21; end; :: RAT is dense, that is there exists at least one rational number between :: any two distinct real numbers. theorem a<b implies ex p st a<p & p<b proof assume A1:a<b; set d=b-a;set m=[\d"/]+1;set l=[\m*a/]+1; a-a<b-a by A1,REAL_1:54; then a+ -a<d by XCMPLX_0:def 8; then A2:0<d by XCMPLX_0:def 6; then A3:0<d" by REAL_1:72; A4: d"<=m by INT_1:52;then A5:0<m by A3; A6:0<>m by A2,A4,REAL_1:72; reconsider p=l/m as Rational; take p; A7:0<m" by A5,REAL_1:72; thus a<p proof m*a-1<[\m*a/] by INT_1:def 4; then m*a<[\m*a/]+1 by REAL_1:84; then m"*(m*a)<m"*l by A7,REAL_1:70; then (m"*m)*a<m"*l by XCMPLX_1:4; then 1*a<m"*l by A6,XCMPLX_0:def 7; hence thesis by XCMPLX_0:def 9; end; A8:[\m*a/]<=m*a by INT_1:def 4; thus p<b proof d"-1<[\d"/] by INT_1:def 4; then d"<[\d"/]+1 by REAL_1:84; then d"*d<m*d by A2,REAL_1:70; then 1<m*d by A2,XCMPLX_0:def 7; then [\m*a/]+1<m*a+m*d by A8,REAL_1:67; then l<m*(a+d) by XCMPLX_1:8; then l<m*(a+(-a+b)) by XCMPLX_0:def 8; then l<m*((a+ -a)+b) by XCMPLX_1:1; then l<m*(0+b) by XCMPLX_0:def 6; then m"*l<m"*(m*b) by A7,REAL_1:70; then m"*l<(m"*m)*b by XCMPLX_1:4; then m"*l<1*b by A6,XCMPLX_0:def 7; hence thesis by XCMPLX_0:def 9; end; end; canceled; theorem Th24: ex m,k st k<>0 & p=m/k proof consider m,n such that A1:n<>0 & p=m/n by Th3; now per cases by A1; suppose 0<n; then n is Nat by INT_1:16; hence thesis by A1; suppose n<0; then 0<=-n by REAL_1:66; then A2:-n is Nat by INT_1:16; A3:p = -(-m/n) by A1 .= -(-m)/n by XCMPLX_1:188 .= (-m)/(-n) by XCMPLX_1:189; -n<>0 by A1,XCMPLX_1:135; hence thesis by A2,A3; end; hence thesis; end; :: Each rational number can be uniquely expressed as a ratio of two :: relatively prime numbers, the first is integer and the latter is natural :: (but not equal to zero). Function denominator(p) is defined as the least :: natural denominator of all denominators of fractions integer/natural=p. :: Function numerator(p) is defined as a product of p and denominator(p). theorem Th25: ex m,k st k<>0 & p=m/k & for n,l st l<>0 & p=n/l holds k<=l proof defpred P[Nat] means ($1<>0 & ex m1 st p=m1/$1); consider m,k such that A1:k<>0 & p=m/k by Th24; A2:ex l st P[l] by A1; ex k1 st P[k1] & for l1 st P[l1] holds k1<=l1 from Min(A2); then consider k1 such that A3: (k1<>0 & ex m1 st p=m1/k1) & for l1 st (l1<>0 & ex n1 st p=n1/l1) holds k1<=l1; consider m1 such that A4:p=m1/k1 & for l1 st (l1<>0 & ex n1 st p=n1/l1) holds k1<=l1 by A3; take m1; take k1; thus k1<>0 by A3; thus m1/k1=p by A4; thus for n,l st l<> 0 & p=n/l holds k1 <= l by A4; thus thesis; end; definition let p; func denominator(p) -> Nat means :Def3:it<>0 & (ex m st p=m/it) & for n,k st k<>0 & p=n/k holds it<=k; existence proof consider m,k such that A1:k<>0 & p=m/k & for n,l st l<>0 & p=n/l holds k<=l by Th25; take k; thus thesis by A1; end; uniqueness proof let k,l such that A2:k<>0 & (ex m st p=m/k) & for m1,k1 st k1<>0 & p=m1/k1 holds k<=k1 and A3:l<>0 & (ex n st p=n/l) & for n1,l1 st l1<>0 & p=n1/l1 holds l<=l1; k<=l & l<=k by A2,A3; hence thesis by AXIOMS:21; end; end; definition let p; func numerator(p) -> Integer equals :Def4: denominator(p)*p; coherence proof consider m such that A1:p=m/denominator(p) by Def3; denominator(p)<>0 by Def3; hence thesis by A1,XCMPLX_1:88; end; end; :: Some basic theorems concerning p, numerator(p) and denominator(p). canceled; theorem Th27: 0<denominator(p) proof 0<>denominator(p) by Def3; hence thesis by NAT_1:19; end; canceled; theorem Th29: 1<=denominator(p) proof 0<denominator(p) by Th27; then 0+1<=denominator(p) by NAT_1:38; hence thesis; end; theorem Th30: 0<denominator(p)" proof 0<denominator(p) by Th27; hence thesis by REAL_1:72; end; canceled 3; theorem Th34: 1>=denominator(p)" proof A1:0<>denominator(p) by Def3; A2:1<=denominator(p) by Th29; 0<=denominator(p)" by Th30; then 1*denominator(p)"<=denominator(p)*denominator(p)" by A2,AXIOMS:25; hence thesis by A1,XCMPLX_0:def 7; end; canceled; theorem Th36: numerator(p)=0 iff p=0 proof A1:numerator(p)=denominator(p)*p by Def4; denominator(p)<>0 by Def3; hence thesis by A1,XCMPLX_1:6; end; theorem Th37: p=numerator(p)/denominator(p) & p=numerator(p)*denominator(p)" & p=denominator(p)"*numerator(p) proof set x=denominator(p);A1:x<>0 by Def3; thus A2:numerator(p)/denominator(p)=numerator(p)*x" by XCMPLX_0:def 9 .=(p*x)*x" by Def4 .=p*(x*x") by XCMPLX_1:4 .=p*1 by A1,XCMPLX_0:def 7 .=p; hence p=numerator(p)*denominator(p)" by XCMPLX_0:def 9; thus p=denominator(p)"*numerator(p) by A2,XCMPLX_0:def 9; end; theorem p<>0 implies denominator(p)=numerator(p)/p proof assume A1:p<>0; A2:denominator(p)<>0 by Def3; p*denominator(p)=numerator(p)/denominator(p)*denominator(p) by Th37; then p"*(p*denominator(p))=p"*numerator(p) by A2,XCMPLX_1:88; then (p"*p)*denominator(p)=p"*numerator(p) by XCMPLX_1:4; then 1*denominator(p)=p"*numerator(p) by A1,XCMPLX_0:def 7; hence denominator(p) =numerator(p)/p by XCMPLX_0:def 9; end; canceled; theorem Th40: p is Integer implies denominator(p)=1 & numerator(p)=p proof assume p is Integer; then reconsider m=p as Integer; p =m/1; then A1:denominator(p)<=1 by Def3; 1<=denominator(p) by Th29; hence A2:denominator(p)=1 by A1,AXIOMS:21; numerator(p)=denominator(p)*p by Def4; hence numerator(p)=p by A2; end; theorem Th41: (numerator(p)=p or denominator(p)=1) implies p is Integer proof assume A1:numerator(p)=p or denominator(p)=1; now per cases by A1; suppose numerator(p)=p;hence thesis; suppose denominator(p)=1; then numerator(p)=p*1 by Def4 .=p; hence thesis; end; hence thesis; end; theorem numerator(p)=p iff denominator(p)=1 proof now assume denominator(p)=1; then p is Integer by Th41; hence numerator(p)=p by Th40; end; hence thesis by Th40; end; canceled; theorem (numerator(p)=p or denominator(p)=1) & 0<=p implies p is Nat proof assume A1:(numerator(p)=p or denominator(p)=1) & 0<=p; then p is Integer by Th41; hence thesis by A1,INT_1:16; end; theorem 1<denominator(p) iff p is not integer proof now assume not p is Integer; then A1:denominator(p)<>1 by Th41; denominator(p)>=1 by Th29; hence denominator(p)>1 by A1,REAL_1:def 5; end; hence thesis by Th40; end; Lm2:1"=1; theorem 1>denominator(p)" iff p is not integer proof A1:denominator(p)"<=1 by Th34; now assume not p is Integer; then denominator(p)" <> 1 by Lm2,Th41; hence denominator(p)"<1 by A1,REAL_1:def 5; end; hence thesis by Lm2,Th40; end; theorem Th47: numerator(p)=denominator(p) iff p=1 proof A1:denominator(p)<>0 by Def3; A2:now assume numerator(p)=denominator(p); then numerator(p)/denominator(p)=1 by A1,XCMPLX_1:60; hence p=1 by Th37; end; now assume A3:p=1; then numerator(p)=1 by Th40; hence numerator(p)=denominator(p) by A3,Th40; end; hence thesis by A2; end; theorem Th48: numerator(p)=-denominator(p) iff p=-1 proof A1:0<>denominator(p) by Def3; A2:now assume numerator(p)=-denominator(p); hence p=(-denominator(p))/denominator(p) by Th37 .=-denominator(p)/denominator(p) by XCMPLX_1:188 .=-1 by A1,XCMPLX_1:60; end; now assume A3:p=-1; then numerator(p)=-1 by Th40; hence numerator(p)=-denominator(p) by A3,Th40; end; hence thesis by A2; end; theorem -numerator(p)=denominator(p) iff p=-1 proof -numerator(p)=denominator(p) iff numerator(p)=-denominator(p); hence thesis by Th48; end; :: We can multiple the numerator and the denominator of any rational number :: by any integer (natural) number which is not equal to zero. theorem Th50: m<>0 implies p=(numerator(p)*m)/(denominator(p)*m) proof assume m<>0; then numerator(p)/denominator(p)=(numerator(p)*m)/(denominator(p)*m) by XCMPLX_1:92; hence p=(numerator(p)*m)/(denominator(p)*m) by Th37; end; canceled 9; theorem Th60: k<>0 & p=m/k implies (ex l st m=numerator(p)*l & k=denominator(p)*l) proof assume A1:k<>0 & p=m/k; A2:denominator(p)<>0 by Def3; defpred P[Nat] means $1*denominator(p)<=k; A3:for l st P[l] holds l<=k proof let l such that A4:l*denominator(p)<=k; assume A5: not thesis; 0<denominator(p) by A2,NAT_1:19; then 0+1<=denominator(p) by NAT_1:38; then A6:k*1<=k*denominator(p) by NAT_1:20; 0<denominator(p) by A2,NAT_1:19; then k*denominator(p)<l*denominator(p) by A5,REAL_1:70; hence contradiction by A4,A6,AXIOMS:22; end; A7:1*denominator(p)<=k by A1,Def3; then A8:ex l1 st P[l1]; consider l such that A9:P[l] and A10:for l1 st P[l1] holds l1<=l from Max(A3,A8); take l; A11:0<>l by A7,A10; then A12:l*denominator(p)<>0 by A2,XCMPLX_1:6; A13: now assume A14:l*denominator(p)<k; then A15: 0+l*denominator(p)<k; then 0<=k-l*denominator(p) by REAL_1:86; then reconsider x=k-l*denominator(p) as Nat by INT_1:16; A16:0<>x by A15,REAL_1:86; m/k=(numerator(p)*l)/(l*denominator(p)) by A1,A11,Th50; then p=(m-numerator(p)*l)/x by A1,A12,A14,XCMPLX_1:124; then denominator(p)<=k-l*denominator(p) by A16,Def3; then l*denominator(p)+1*denominator(p)<=k by REAL_1:84; then (l+1)*denominator(p)<=k by XCMPLX_1:8; then l+1<=l by A10; hence contradiction by NAT_1:38; end; then A17:k=denominator(p)*l by A9,AXIOMS:21; p*k=m by A1,XCMPLX_1:88; hence m=numerator(p)/denominator(p)*(denominator(p)*l) by A17,Th37 .=(numerator(p)/denominator(p)*denominator(p))*l by XCMPLX_1:4 .=numerator(p)*l by A2,XCMPLX_1:88; thus k=denominator(p)*l by A9,A13,AXIOMS:21; end; theorem p=m/n & n<>0 implies ex m1 st m=numerator(p)*m1 & n=denominator(p)*m1 proof assume A1:p=m/n & n<>0; per cases by A1; suppose n<0; then 0<=-n by REAL_1:66; then reconsider x=-n as Nat by INT_1:16;A2:x<>0 by A1,XCMPLX_1:135; p=-(-m/n) by A1 .=-(-m)/n by XCMPLX_1:188 .=(-m)/x by XCMPLX_1:189; then consider k such that A3:-m=numerator(p)*k & x=denominator(p)*k by A2,Th60; take y=-k; thus m=-numerator(p)*k by A3 .=numerator(p)*y by XCMPLX_1:175; thus n=-denominator(p)*k by A3 .=denominator(p)*y by XCMPLX_1:175; suppose 0<n; then reconsider x=n as Nat by INT_1:16; consider k such that A4:m=numerator(p)*k & x=denominator(p)*k by A1,Th60; reconsider y=k as Integer;take y; thus m=numerator(p)*y by A4; thus n=denominator(p)*y by A4; end; :: Fraction numerator(p)/denominator(p) is irreducible. theorem Th62: not ex l st 1<l & ex m,k st numerator(p)=m*l & denominator(p)=k*l proof assume not thesis; then consider l such that A1:1<l & ex m,k st numerator(p)=m*l & denominator(p)=k*l; consider m,k such that A2:numerator(p)=m*l & denominator(p)=k*l by A1; A3:0<>l by A1; A4:k<>0 by A2,Def3; then A5:0<k by NAT_1:19; A6:now k*1<k*l by A1,A5,REAL_1:70; hence k<denominator(p) by A2; end; p=(m*l)/(k*l) by A2,Th37 .=(m/k)*(l/l) by XCMPLX_1:77 .=m/k*1 by A3,XCMPLX_1:60 .=m/k; hence contradiction by A4,A6,Def3; end; theorem Th63: (p=m/k & k<>0 & not ex l st 1<l & ex m1,k1 st m=m1*l & k=k1*l) implies k=denominator(p) & m=numerator(p) proof assume A1:(p=m/k & k<>0 & not ex l st 1<l & ex m1,k1 st m=m1*l & k=k1*l); then consider l such that A2:m=numerator(p)*l & k=denominator(p)*l by Th60; A3:l<=1 by A1,A2; l<>0 by A1,A2; then 0<l by NAT_1:19; then 0+1<=l by NAT_1:38; then A4:l=1 by A3,AXIOMS:21; hence k=denominator(p) by A2; thus m=numerator(p) by A2,A4; end; theorem Th64: p<-1 iff numerator(p)<-denominator(p) proof A1:0<>denominator(p) by Def3; A2:0<denominator(p)" by Th30; A3:0<denominator(p) by Th27; A4:now assume p<-1; then numerator(p)/denominator(p)<-1 by Th37; then (numerator(p)/denominator(p))*denominator(p)<(-1)*denominator(p) by A3,REAL_1:70; then numerator(p)<(-1)*denominator(p) by A1,XCMPLX_1:88; then numerator(p)<-(1*denominator(p)) by XCMPLX_1:175; hence numerator(p)<-denominator(p); end; now assume numerator(p)<-denominator(p); then numerator(p)<-(1*denominator(p)); then numerator(p)<(-1)*denominator(p) by XCMPLX_1:175; then numerator(p)*denominator(p)"<((-1)*denominator(p))*denominator(p)" by A2,REAL_1:70; then numerator(p)*denominator(p)"<(-1)*(denominator(p)*denominator(p)") by XCMPLX_1:4; then numerator(p)*denominator(p)"<(-1)*1 by A1,XCMPLX_0:def 7; hence p<-1 by Th37; end; hence thesis by A4; end; theorem Th65: p<=-1 iff numerator(p)<=-denominator(p) proof A1:now assume A2:p<=-1; now per cases by A2,REAL_1:def 5; suppose p=-1; hence numerator(p)<=-denominator(p) by Th48; suppose p<-1; hence numerator(p)<=-denominator(p) by Th64; end; hence numerator(p)<=-denominator(p); end; now assume A3:numerator(p)<=-denominator(p); now per cases by A3,REAL_1:def 5; suppose numerator(p)=-denominator(p); hence p<=-1 by Th48; suppose numerator(p)<-denominator(p); hence p<=-1 by Th64; end; hence p<=-1; end; hence thesis by A1; end; theorem p<-1 iff denominator(p)<-numerator(p) proof denominator(p)<-numerator(p) iff -(-numerator(p))<-denominator(p) by REAL_1:50; hence thesis by Th64; end; theorem p<=-1 iff denominator(p)<=-numerator(p) proof denominator(p)<=-numerator(p) iff -(-numerator(p))<=-denominator(p) by REAL_1:50; hence thesis by Th65; end; canceled 4; theorem Th72: p<1 iff numerator(p)<denominator(p) proof A1:0<denominator(p) by Th27; A2:0<>denominator(p) by Def3; A3:0<denominator(p)" by Th30; A4:now assume p<1; then numerator(p)/denominator(p)<1 by Th37; then numerator(p)/denominator(p)*denominator(p)<1*denominator(p) by A1,REAL_1:70; hence numerator(p)<denominator(p) by A2,XCMPLX_1:88; end; now assume numerator(p)<denominator(p); then numerator(p)*denominator(p)"<denominator(p)*denominator(p)" by A3,REAL_1:70; then numerator(p)*denominator(p)"<1 by A2,XCMPLX_0:def 7; hence p<1 by Th37; end; hence thesis by A4; end; theorem p<=1 iff numerator(p)<=denominator(p) proof A1:now assume A2:p<=1; now per cases by A2,REAL_1:def 5; suppose p=1; hence numerator(p)<=denominator(p) by Th47; suppose p<1; hence numerator(p)<=denominator(p) by Th72; end; hence numerator(p)<=denominator(p); end; now assume A3:numerator(p)<=denominator(p); now per cases by A3,REAL_1:def 5; suppose numerator(p)=denominator(p); hence p<=1 by Th47; suppose numerator(p)<denominator(p); hence p<=1 by Th72; end; hence p<=1; end; hence thesis by A1; end; canceled 2; theorem Th76: p<0 iff numerator(p)<0 proof A1:0<denominator(p) by Th27; A2:0<>denominator(p) by Def3; A3:0<denominator(p)" by Th30; A4:now assume p<0; then numerator(p)/denominator(p)<0 by Th37; then numerator(p)/denominator(p)*denominator(p)<0*denominator(p) by A1,REAL_1:70; hence numerator(p)<0 by A2,XCMPLX_1:88; end; now assume numerator(p)<0; then numerator(p)*denominator(p)"<0*denominator(p)" by A3,REAL_1:70; hence p<0 by Th37; end; hence thesis by A4; end; theorem Th77: p<=0 iff numerator(p)<=0 proof A1:now assume A2:p<=0; now per cases by A2; suppose p=0; hence numerator(p)<=0 by Th40; suppose p<0; hence numerator(p)<=0 by Th76; end; hence numerator(p)<=0; end; now assume A3:numerator(p)<=0; now per cases by A3; suppose numerator(p)=0; hence p<=0 by Th36; suppose numerator(p)<0; hence p<=0 by Th76; end; hence p<=0; end; hence thesis by A1; end; canceled 2; theorem Th80: a<p iff a*denominator(p)<numerator(p) proof A1:0<denominator(p) by Th27; A2:0<>denominator(p) by Def3; A3:0<denominator(p)" by Th30; A4:now assume a<p; then a*denominator(p)<p*denominator(p) by A1,REAL_1:70; hence a*denominator(p)<numerator(p) by Def4; end; now assume a*denominator(p)<numerator(p); then (a*denominator(p))*denominator(p)"<numerator(p)*denominator(p)" by A3,REAL_1:70; then a*(denominator(p)*denominator(p)")<numerator(p)*denominator(p)" by XCMPLX_1:4; then a*1<numerator(p)*denominator(p)" by A2,XCMPLX_0:def 7; hence a<p by Th37; end; hence thesis by A4; end; theorem a<=p iff a*denominator(p)<=numerator(p) proof A1:denominator(p)<>0 by Def3; A2:now assume A3:a<=p; now per cases by A3,REAL_1:def 5; suppose a=p; hence numerator(p)>=a*denominator(p) by Def4; suppose a<p; hence a*denominator(p)<=numerator(p) by Th80; end; hence a*denominator(p)<=numerator(p); end; now assume A4:a*denominator(p)<=numerator(p); now per cases by A4,REAL_1:def 5; suppose a*denominator(p)=numerator(p); then p=(a*denominator(p))/(1*denominator(p)) by Th37 .=(a/1)*(denominator(p)/denominator(p)) by XCMPLX_1:77 .=a/1*1 by A1,XCMPLX_1:60 .=a; hence a<=p; suppose a*denominator(p)<numerator(p); hence a<=p by Th80; end; hence a<=p; end; hence thesis by A2; end; canceled 2; theorem p=q iff (denominator(p)=denominator(q) & numerator(p)=numerator(q)) proof now assume denominator(p)=denominator(q) & numerator(p)=numerator(q); hence p=numerator(q)/denominator(q) by Th37 .=q by Th37; end; hence thesis; end; canceled; theorem p<q iff numerator(p)*denominator(q)<numerator(q)*denominator(p) proof A1:0<denominator(p) & 0<denominator(q) by Th27; p<q iff numerator(p)/denominator(p)<q by Th37; then p<q iff numerator(p)/denominator(p)<numerator(q)/denominator(q) by Th37; hence thesis by A1,Lm1; end; theorem Th87: denominator(-p)=denominator(p) & numerator(-p)=-numerator(p) proof A1:denominator(p)<>0 by Def3; A2:denominator(-p)<>0 by Def3; A3:0<=denominator(p) by Th27; A4:0<=denominator(-p) by Th27; A5:p=numerator(p)/denominator(p) by Th37; -p=-numerator(p)/denominator(p) by Th37 .=(-numerator(p))/denominator(p) by XCMPLX_1:188; then consider k such that A6:-numerator(p)=numerator(-p)*k & denominator(p)=denominator(-p)*k by A1,Th60; k<>0 by A6,Def3; then 0<k by NAT_1:19; then 0+1<=k by NAT_1:38; then A7: denominator(-p)*1<=denominator(-p)*k by A4,AXIOMS:25; A8:p=-(-p) .=-numerator(-p)/denominator(-p) by Th37 .=(-numerator(-p))/denominator(-p) by XCMPLX_1:188; then consider l such that A9:-numerator(-p)=numerator(p)*l & denominator(-p)=denominator(p)*l by A2,Th60; l<>0 by A9,Def3; then 0<l by NAT_1:19; then 0+1<=l by NAT_1:38; then A10: denominator(p)*1<=denominator(p)*l by A3,AXIOMS:25; hence denominator(p)=denominator(-p) by A6,A7,A9,AXIOMS:21; numerator(p)/denominator(p)*denominator(p) =(-numerator(-p))/denominator(p)*denominator(p) by A5,A6,A7,A8,A9,A10,AXIOMS:21; then numerator(p)=(-numerator(-p))/denominator(p)*denominator(p) by A1,XCMPLX_1:88; then -numerator(p)=-(-numerator(-p)) by A1,XCMPLX_1:88; hence -numerator(p)=numerator(-p); end; theorem Th88: 0<p & q=1/p iff numerator(q)=denominator(p) & denominator(q)=numerator(p) proof A1:now assume A2:0<p & q=1/p; then A3:0<numerator(p) by Th77;A4:0<>numerator(p) by A2,Th77; A5:q=1/(numerator(p)/denominator(p)) by A2,Th37 .=(1*denominator(p))/numerator(p) by XCMPLX_1:78 .=denominator(p)/numerator(p); reconsider x=denominator(p) as Integer; reconsider y=numerator(p) as Nat by A3,INT_1:16; not ex k st 1<k & ex m,l st x=m*k & y=l*k proof assume not thesis; then consider k such that A6:1<k & ex m,l st x=m*k & y=l*k; consider m,l such that A7:x=m*k & y=l*k by A6; A8: 0<k by A6,AXIOMS:22; now assume not 0<=m; then m*k<0*m by A8,REAL_1:71; hence contradiction by A7,Th27; end; then reconsider z=m as Nat by INT_1:16; reconsider v=l as Integer; 1<k & numerator(p)=v*k & denominator(p)=z*k by A6,A7; hence contradiction by Th62; end; hence numerator(p)=denominator(q) & denominator(p)=numerator(q) by A4,A5, Th63; end; now assume A9:numerator(q)=denominator(p) & denominator(q)=numerator(p); then 0<numerator(p) by Th27; hence 0<p by Th77; thus q =(1*denominator(p))/numerator(p) by A9,Th37 .=1/(numerator(p)/denominator(p)) by XCMPLX_1:78 .=1/p by Th37; end; hence thesis by A1; end; theorem p<0 & q=1/p iff numerator(q)=-denominator(p) & denominator(q)=-numerator(p) proof A1:now assume A2:p<0 & q=1/p; set r=-p;A3:0<r by A2,REAL_1:66; set s=-q; s=1/r by A2,XCMPLX_1:189; then numerator(s)=denominator(r) & denominator(s)=numerator(r) by A3,Th88; then -numerator(q)=denominator(r) & denominator(q)=numerator(r) by Th87; then -(-numerator(q))=-denominator(p) & denominator(q)=-numerator(p) by Th87; hence numerator(q)=-denominator(p) & denominator(q)=-numerator(p); end; now assume A4:numerator(q)=-denominator(p) & denominator(q)=-numerator(p); then 0<-numerator(p) by Th27; then numerator(p)<0 by REAL_1:66; hence p<0 by Th76; thus q=(-denominator(p))/(-numerator(p)) by A4,Th37 .=-(denominator(p)/(-numerator(p))) by XCMPLX_1:188 .=-(-denominator(p)/numerator(p)) by XCMPLX_1:189 .=(1*denominator(p))/numerator(p) .=1/(numerator(p)/denominator(p)) by XCMPLX_1:78 .=1/p by Th37; end; hence thesis by A1; end;

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