
theorem Th8:
  for f1,f2,f3 being Element of REAL_Music st
  (the Ratio of REAL_Music).(f1,f2) = (the Ratio of REAL_Music).(f1,f3)
  holds f2 = f3
  proof
    set S = REAL_Music;
    let f1,f2,f3 be Element of S;
    assume
A1: (the Ratio of S).(f1,f2) = (the Ratio of S).(f1,f3);
    reconsider x = [f1,f2],y = [f1,f3] as Element of [:REALPLUS,REALPLUS:]
      by ZFMISC_1:def 2;
    consider y9,z9 be Element of REALPLUS such that
A2: x = [y9,z9] and
A3: REAL_ratio.x = REAL_ratio(y9,z9) by Def02;
    consider y99,z99 be Element of REALPLUS such that
A4: y = [y99,z99] and
A5: REAL_ratio.y = REAL_ratio(y99,z99) by Def02;
    consider r,s be positive Real such that
A6: y9 = r & s = z9 & REAL_ratio(y9,z9) = s / r by Def01;
    consider r9,s9 be positive Real such that
A7: y99 = r9 & s9 = z99 & REAL_ratio(y99,z99) = s9 / r9 by Def01;
A8: y9 = f1 & z9 = f2 & y99 = f1 & z99 = f3 by A2,A4,XTUPLE_0:1;
    REAL_ratio.(f1,f2) = REAL_ratio(y9,z9) &
      REAL_ratio.(f1,f3) = REAL_ratio(y99,z99) by A5,A3,BINOP_1:def 1;
    hence thesis by A6,A7,A8,A1,XCMPLX_1:53;
  end;
