reserve MS for satisfying_equiv MusicStruct;
reserve a,b,c,d,e,f for Element of MS;

theorem
  for MS being satisfying_harmonique_stable satisfying_linearite_harmonique
  satisfying_harmonic_closed satisfying_Nat satisfying_tonic
  satisfying_interval satisfying_equiv MusicStruct
  for frequency being Element of MS holds octave(MS,frequency) = octave(MS)
  proof
    let MS be satisfying_harmonique_stable satisfying_linearite_harmonique
    satisfying_harmonic_closed satisfying_Nat satisfying_tonic
    satisfying_interval satisfying_equiv MusicStruct;
    let frequency be Element of MS;
A1: now
      let x be object;
      assume
A2:   x in octave(MS,frequency);
      then consider x1,x2 be object such that
A3:   x1 in the carrier of MS and
A4:   x2 in the carrier of MS and
A5:   x = [x1,x2] by ZFMISC_1:def 2;
      reconsider x1,x2 as Element of MS by A3,A4;
A6:   NATPLUS c= the carrier of MS by Def12a;
      1 in NATPLUS & 2 in NATPLUS by NAT_LAT:def 6;
      then reconsider y1 = 1,y2 = 2 as Element of MS by A6;
      set z = 1-harmonique(MS,frequency),
      t = 2-harmonique(MS,frequency);
      reconsider n = 1,m = 2 as non zero Nat;
      consider r2 be positive Real such that
A7:   y1 = r2 and
A8:   m-harmonique(MS,y1) = m * r2 by Def09;
      set a = n-harmonique(MS,frequency), b = m-harmonique(MS,frequency);
A9:   y1 = n-harmonique(MS,y1) by Th40;
A10:  a,b equiv x1,x2 by A2,A5,EQREL_1:18;
      y1,y2 equiv a,b by A9,A7,A8,Def10;
      then y1,y2 equiv x1,x2 by A10,Th23;
      hence x in octave(MS) by A5,EQREL_1:18;
    end;
    now
      let x be object;
      assume
A11:  x in octave(MS);
      then consider x1,x2 be object such that
A12:  x1 in the carrier of MS and
A13:  x2 in the carrier of MS and
A14:  x = [x1,x2] by ZFMISC_1:def 2;
      reconsider x1,x2 as Element of MS by A12,A13;
A15:  NATPLUS c= the carrier of MS by Def12a;
      1 in NATPLUS & 2 in NATPLUS by NAT_LAT:def 6;
      then reconsider y = 1,z = 2 as Element of MS by A15;
      reconsider y9 = y as positive Real;
      reconsider n = 1,m = 2 as non zero Nat;
      set a = n-harmonique(MS,frequency), b = m-harmonique(MS,frequency),
      c = n-harmonique(MS,y), d = m-harmonique(MS,y);
A16:  a,b equiv c,d by Def10;
      reconsider n1 = 1,n2 = 2 as Element of MS by Th20;
      consider r1 be positive Real such that
A17:  n1 = r1 and
A18:  n-harmonique(MS,n1) = n * r1 by Def09;
      consider r2 be positive Real such that
A19:  n1 = r2 and
A20:  m-harmonique(MS,n1) = m * r2 by Def09;
      n1,n2 equiv x1,x2 by A11,A14,EQREL_1:18;
      then a,b equiv x1,x2 by A16,A17,A18,A19,A20,Th23;
      hence x in octave(MS,frequency) by A14,EQREL_1:18;
    end;
    hence thesis by A1;
  end;
