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

theorem Th32:
  REAL_Music is satisfying_harmonique_stable
  proof
    set MS = REAL_Music;
    let f1,f2 be Element of MS;
    let n,m be non zero Nat;
    reconsider r1 = f1,r2 = f2 as positive Real by Th1;
    (ex fr1 be positive Real st fr1 = f1 &
      n-harmonique(MS,f1) = n * fr1) &
      (ex fr2 be positive Real st fr2 = f1 &
      m-harmonique(MS,f1) = m * fr2) &
      (ex fr3 be positive Real st fr3 = f2 &
      n-harmonique(MS,f2) = n * fr3) &
      (ex fr4 be positive Real st fr4 = f2 &
      m-harmonique(MS,f2) = m * fr4) by Def09;
    hence n-harmonique(MS,f1),m-harmonique(MS,f1) equiv
      n-harmonique(MS,f2),m-harmonique(MS,f2) by Th11;
  end;
