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

theorem Th33:
  RAT_Music is satisfying_harmonique_stable
  proof
    set MS = RAT_Music;
    now
      let f1,f2 be Element of MS;
      let n,m be non zero Nat;
      set fn1 = n-harmonique(MS,f1), fm1 = m-harmonique(MS,f1),
      fn2 = n-harmonique(MS,f2), fm2 = m-harmonique(MS,f2);
      reconsider r1 = f1,r2 = f2 as positive Rational by Th2;
      consider fr1 be positive Real such that
A1:   fr1 = f1 and
A2:   n-harmonique(MS,f1) = n * fr1 by Def09;
      consider fr2 be positive Real such that
A3:   fr2 = f1 and
A4:   m-harmonique(MS,f1) = m * fr2 by Def09;
      consider fr3 be positive Real such that
A5:   fr3 = f2 and
A6:   n-harmonique(MS,f2) = n * fr3 by Def09;
      consider fr4 be positive Real such that
A7:   fr4 = f2 and
A8:   m-harmonique(MS,f2) = m * fr4 by Def09;
      fn1 = n * r1 & fm1 = m * r1 & fn2 = n * r2 & fm2 = m * r2
        by A1,A2,A3,A4,A5,A6,A7,A8;
      hence n-harmonique(MS,f1),m-harmonique(MS,f1) equiv
        n-harmonique(MS,f2),m-harmonique(MS,f2) by Th18;
    end;
    hence thesis;
  end;
