reserve MS for satisfying_equiv MusicStruct;
reserve a,b,c,d,e,f for Element of MS;
reserve MS for satisfying_octave_descendent_constructible
  classical_octave satisfying_octave_constructible classical_fifth
  satisfying_fifth_constructible satisfying_harmonic_closed satisfying_Nat
  satisfying_commutativity satisfying_interval satisfying_equiv satisfying_Real
  non empty MusicStruct,
  fondamentale,frequency for Element of MS;
reserve                              MS for MusicSpace,
        fondamentale, frequency, f1, f2 for Element of MS;

theorem Th78:
  for MS being MusicSpace st MS = REAL_Music holds
  for frequency being Element of MS holds
  ex fr,qr being positive Real st
  fr = frequency & qr = (4 qua Real) / 3 * fr &
  [fr,qr] in fourth(MS)
  proof
    let MS be MusicSpace;
    assume
A1: MS = REAL_Music;
    now
      let frequency be Element of MS;
      reconsider f = frequency as positive Real by A1,Th1;
      reconsider qr = (4 qua Real) / 3 * f as positive Real;
      reconsider q = qr as Element of MS by A1,Th1;
      take f,qr;
      thus f = frequency;
      thus qr = (4 qua Real) / 3 * f;
      reconsider n2 = 3,n3 = 4 as Element of MS by Th20;
      reconsider x = [n2,n3],y = [frequency,q] as
        Element of [:REALPLUS,REALPLUS:] by A1;
      reconsider z = [frequency,q] as Element of [:REALPLUS,REALPLUS:]
        by A1;
      consider x9,y9 be Element of REALPLUS such that
A2:   z = [x9,y9] and
A3:   REAL_ratio.z = REAL_ratio(x9,y9) by Def02;
      consider r,s be positive Real such that
A4:   x9 = r & s = y9 & REAL_ratio(x9,y9) = s / r by Def01;
      consider x99,y99 be Element of REALPLUS such that
A5:   x = [x99,y99] and
A6:   REAL_ratio.x = REAL_ratio(x99,y99) by Def02;
      consider r9,s9 be positive Real such that
A7:   x99 = r9 & s9 = y99 & REAL_ratio(x99,y99) = s9 / r9 by Def01;
A8:   n2 = r9 & n3 = s9 & r = frequency & s = q
        by A4,A2,A5,A7,XTUPLE_0:1;
      now
        thus REAL_ratio.(n2,n3) = (4 qua Real) / (3 qua Real)
          by A7,A8,A6,BINOP_1:def 1;
        thus REAL_ratio.(frequency,q) = s / r by A3,A4,BINOP_1:def 1
                                     .= (4 qua Real) / 3 by A8,XCMPLX_1:89;
      end;
      then n2,n3 equiv frequency,q by A1,Def08a;
      hence [f,qr] in fourth(MS) by EQREL_1:18;
    end;
    hence thesis;
  end;
