
theorem Th17:
  for f1,f2,f3 being Element of RAT_Music st
  (the Ratio of RAT_Music).(f1,f1) = (the Ratio of RAT_Music).(f2,f3) holds
 f2 = f3
  proof
    let f1,f2,f3 be Element of RAT_Music;
    assume
A1: (the Ratio of RAT_Music).(f1,f1) =
    (the Ratio of RAT_Music).(f2,f3);
    set S = RAT_Music;
    reconsider x = [f1,f1],y = [f2,f3] as Element of [:RATPLUS,RATPLUS:]
      by ZFMISC_1:def 2;
    consider y9,z9 be Element of RATPLUS such that
A2: x = [y9,z9] and
A3: RAT_ratio.x = RAT_ratio(y9,z9) by Def05;
    consider y99,z99 be Element of RATPLUS such that
A4: y = [y99,z99] and
A5: RAT_ratio.y = RAT_ratio(y99,z99) by Def05;
    consider r,s be positive Rational such that
A6: y9 = r & s = z9 & RAT_ratio(y9,z9) = s / r by Def04;
    consider r9,s9 be positive Rational such that
A7: y99 = r9 & s9 = z99 & RAT_ratio(y99,z99) = s9 / r9 by Def04;
A8: y9 = f1 & z9 = f1 & y99 = f2 & z99 = f3 by A2,A4,XTUPLE_0:1;
    RAT_ratio.(f1,f1) = RAT_ratio(y9,z9) &
      RAT_ratio.(f2,f3) = RAT_ratio(y99,z99) by A5,A3,BINOP_1:def 1;
    hence f2 = f3 by A7,A8,XCMPLX_1:58,XCMPLX_1:60,A6,A1;
  end;
