
theorem Th12:
  for f1,f2,f3,f4 be Element of REAL_Music holds
    ((the Ratio of REAL_Music).(f1,f2) = (the Ratio of REAL_Music).(f3,f4)) iff
    (the Ratio of REAL_Music).(f2,f1) = (the Ratio of REAL_Music).(f4,f3)
  proof
    let f1,f2,f3,f4 be Element of REAL_Music;
    set MS = REAL_Music;
    reconsider x = [f1,f2],y = [f3,f4] as Element of [:REALPLUS,REALPLUS:]
      by ZFMISC_1:def 2;
    consider y9,z9 be Element of REALPLUS such that
A1: x = [y9,z9] and
A2: REAL_ratio.x = REAL_ratio(y9,z9) by Def02;
    consider y99,z99 be Element of REALPLUS such that
A3: y = [y99,z99] and
A4: REAL_ratio.y = REAL_ratio(y99,z99) by Def02;
    reconsider x1 = [z9,y9],y1 = [z99,y99] as Element of [:REALPLUS,REALPLUS:];
    consider y19,z19 be Element of REALPLUS such that
A5: x1 = [y19,z19] and
A6: REAL_ratio.x1 = REAL_ratio(y19,z19) by Def02;
    consider y199,z199 be Element of REALPLUS such that
A7: y1 = [y199,z199] and
A8: REAL_ratio.y1 = REAL_ratio(y199,z199) by Def02;
A9: f1 = y9 & f2 = z9 & f3 = y99 & f4 = z99 by A1,A3,XTUPLE_0:1;
    then
A10: f1 = z19 & f2 = y19 & f3 = z199 & f4 = y199 by A5,A7,XTUPLE_0:1;
A11: ((the Ratio of MS).(f2,f1)) = REAL_ratio(y19,z19) &
      ((the Ratio of MS).(f4,f3)) = REAL_ratio(y199,z199)
      by A9,BINOP_1:def 1,A6,A8;
    hereby
      assume
A12:  ((the Ratio of MS).(f1,f2) = (the Ratio of MS).(f3,f4));
      REAL_ratio(y9,z9) = REAL_ratio.(f1,f2) by A2,BINOP_1:def 1
                       .= REAL_ratio(y99,z99) by A12,A4,BINOP_1:def 1;
      hence (the Ratio of MS).(f2,f1) = (the Ratio of MS).(f4,f3)
        by A9,A10,A11,Th6;
    end;
    assume
A13: (the Ratio of MS).(f2,f1) = (the Ratio of MS).(f4,f3);
    REAL_ratio.(f1,f2) = REAL_ratio(z19,y19)
      by BINOP_1:def 1,A9,A10,A2
                      .= REAL_ratio(z199,y199) by A13,A11,Th6
                      .= REAL_ratio.(f3,f4) by BINOP_1:def 1,A9,A10,A4;
    hence ((the Ratio of MS).(f1,f2) = (the Ratio of MS).(f3,f4));
  end;
