
theorem Th7:
  REAL_Music is non empty &
  the carrier of REAL_Music c= REALPLUS &
  (for f1,f2,f3,f4 being Element of REAL_Music holds
    (f1,f2 equiv f3,f4 iff
    (the Ratio of REAL_Music).(f1,f2) = (the Ratio of REAL_Music).(f3,f4)))
  proof
    set T = REAL_Music;
    thus T is non empty;
    thus the carrier of T c= REALPLUS;
    thus for f1,f2,f3,f4 being Element of T holds
      (f1,f2 equiv f3,f4 iff
      (the Ratio of T).(f1,f2) = (the Ratio of T).(f3,f4))
    proof
      let f1,f2,f3,f4 be Element of T;
      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;
      hereby
        assume f1,f2 equiv f3,f4;
        then consider a,b,c,d be Element of REALPLUS such that
A5:     x = [a,b] & y = [c,d] and
A6:     REAL_ratio(a,b) = REAL_ratio(c,d) by Def03;
        y9 = a & z9 = b & y99 = c & z99 = d by A1,A3,A5,XTUPLE_0:1;
        then a = f1 & b = f2 & c = f3 & d = f4 &
          (the Ratio of T).(a,b) = REAL_ratio(a,b) &
          (the Ratio of T).(c,d) = REAL_ratio(c,d)
          by XTUPLE_0:1,A2,A4,A5,BINOP_1:def 1;
        hence (the Ratio of T).(f1,f2) = (the Ratio of T).(f3,f4) by A6;
      end;
      assume
A7:   (the Ratio of T).(f1,f2) = (the Ratio of T).(f3,f4);
      REAL_ratio(y9,z9) = REAL_ratio.(f1,f2) by A2,BINOP_1:def 1
                       .= REAL_ratio(y99,z99) by A7,A4,BINOP_1:def 1;
      hence f1,f2 equiv f3,f4 by A1,A3,Def03;
    end;
  end;
