
theorem Th6:
  for a,b,c,d being Element of REALPLUS holds
  REAL_ratio(a,b) = REAL_ratio(c,d) iff REAL_ratio(b,a) = REAL_ratio(d,c)
  proof
    let a,b,c,d be Element of REALPLUS;
    consider r,s be positive Real such that
A1: a = r & b = s & REAL_ratio(a,b) = s/r by Def01;
    consider r9,s9 be positive Real such that
A2: c = r9 & d = s9 & REAL_ratio(c,d) = s9/r9 by Def01;
    consider r99,s99 be positive Real such that
A3: b = r99 & a = s99 & REAL_ratio(b,a) = s99/r99 by Def01;
    consider r999,s999 be positive Real such that
A4: d = r999 & c = s999 & REAL_ratio(d,c) = s999/r999 by Def01;
    hereby
      assume
A5:   REAL_ratio(a,b) = REAL_ratio(c,d);
      1 / (s / r) = r / s by XCMPLX_1:57;
      then r / s = r9 / s9 by A5,A1,A2,XCMPLX_1:57;
      hence REAL_ratio(b,a) = REAL_ratio(d,c) by A1,A2,A3,Def01;
    end;
    assume
A6: REAL_ratio(b,a) = REAL_ratio(d,c);
    1 / (s99 / r99) = r99 / s99 by XCMPLX_1:57;
    then r99 / s99 = r999 / s999 by A6,A3,A4,XCMPLX_1:57;
    hence REAL_ratio(a,b) = REAL_ratio(c,d) by Def01,A2,A4,A3;
  end;
