
theorem Th13:
  for a,b,c,d being Element of RATPLUS holds
  RAT_ratio(a,b) = RAT_ratio(c,d) iff RAT_ratio(b,a) = RAT_ratio(d,c)
  proof
    let a,b,c,d be Element of RATPLUS;
    consider r,s be positive Rational such that
A1: a = r & b = s & RAT_ratio(a,b) = s/r by Def04;
    consider r9,s9 be positive Rational such that
A2: c = r9 & d = s9 & RAT_ratio(c,d) = s9/r9 by Def04;
    consider r99,s99 be positive Rational such that
A3: b = r99 & a = s99 & RAT_ratio(b,a) = s99/r99 by Def04;
    consider r999,s999 be positive Rational such that
A4: d = r999 & c = s999 & RAT_ratio(d,c) = s999/r999 by Def04;
    hereby
      assume
A5:   RAT_ratio(a,b) = RAT_ratio(c,d);
      1 / (s / r) = r / s by XCMPLX_1:57;
      hence RAT_ratio(b,a) = RAT_ratio(d,c) by A1,A2,A3,A4,A5,XCMPLX_1:57;
    end;
    assume
A6: RAT_ratio(b,a) = RAT_ratio(d,c);
    1 / (s99 / r99) = r99 / s99 by XCMPLX_1:57;
    hence RAT_ratio(a,b) = RAT_ratio(c,d) by A1,A2,A3,A4,A6,XCMPLX_1:57;
  end;
