
theorem Th9:
  for frequency being Element of REAL_Music
  for n being non zero Nat ex harmonique be Element of REAL_Music st
  [frequency,harmonique] in Class(the Equidistance of REAL_Music,[1,n])
  proof
    set S = REAL_Music;
    now
      let frequency be Element of S;
      let n be non zero Nat;
      reconsider f = frequency as positive Real by Th1;
      reconsider harmonique = n * f as Element of S by Th1;
      take harmonique;
      reconsider x = 1,y = n as Element of S by Th1;
      reconsider z = [x,y] as Element of [:REALPLUS,REALPLUS:]
        by ZFMISC_1:def 2;
      consider x9,y9 be Element of REALPLUS such that
A1:   z = [x9,y9] and
A2:   REAL_ratio.z = REAL_ratio(x9,y9) by Def02;
      reconsider z9 = [frequency,harmonique] as
        Element of [:REALPLUS,REALPLUS:] by ZFMISC_1:def 2;
      consider x99,y99 be Element of REALPLUS such that
A3:   z9 = [x99,y99] and
A4:   REAL_ratio.z9 = REAL_ratio(x99,y99) by Def02;
      consider r,s be positive Real such that
A5:   x9 = r & s = y9 & REAL_ratio(x9,y9) = s / r by Def01;
A6:   r = 1 & s = n by A5,A1,XTUPLE_0:1;
      consider r9,s9 be positive Real such that
A7:   x99 = r9 & s9 = y99 & REAL_ratio(x99,y99) = s9 / r9 by Def01;
A8:   r9 = frequency & s9 = harmonique by A7,A3,XTUPLE_0:1;
      now
        thus (the Ratio of S).(x,y) = n by A5,A6,A2,BINOP_1:def 1;
        thus (the Ratio of S).(frequency,harmonique) = REAL_ratio(x99,y99)
          by A4,BINOP_1:def 1
                                                    .= n by A7,A8,XCMPLX_1:89;
      end;
      then x,y equiv frequency,harmonique by Th7;
      hence [frequency,harmonique] in Class(the Equidistance of S,[1,n])
        by EQREL_1:18;
    end;
    hence thesis;
  end;
