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

theorem Th30:
  REAL_Music is satisfying_linearite_harmonique
  proof
    set MS = REAL_Music;
    now
      let frequency be Element of MS;
      let n be non zero Nat;
      reconsider fr = frequency as positive Real by Th1;
      take fr;
      thus frequency = fr;
      reconsider n1 = 1, n2 = n as Element of MS by Th20;
      reconsider f2 = n * fr as Element of MS by Th1;
      reconsider x = [n1,n2], y= [frequency,f2] as
        Element of [:REALPLUS,REALPLUS:];
      now
        thus [frequency,n-harmonique(MS,frequency)] in
          Class(the Equidistance of MS,[1,n]) by Def08b;
        now
          thus (the Ratio of MS).(n1,n2) = REAL_ratio.(n1,n2);
          thus (the Ratio of MS).(frequency,f2) = REAL_ratio.(frequency,f2);
          now
            reconsider n19 = n1,n29 = n2,fr9 = fr, f29 = f2 as
              Element of REALPLUS;
            consider r,s be positive Real such that
A1:         n19 = r & n29 = s & REAL_ratio(n19,n29) = s / r by Def01;
            consider r9,s9 be positive Real such that
A2:         fr9 = r9 & f29 = s9 & REAL_ratio(fr9,f29) = s9 / r9 by Def01;
            consider y9,z9 be Element of REALPLUS such that
A3:         x = [y9,z9] and
A4:         REAL_ratio.x = REAL_ratio(y9,z9) by Def02;
            consider y99,z99 be Element of REALPLUS such that
A5:         y = [y99,z99] and
A6:         REAL_ratio.y = REAL_ratio(y99,z99) by Def02;
A7:         y9 = n1 & z9 = n2 & y99 = frequency & z99 = f2
              by A3,A5,XTUPLE_0:1;
            hence REAL_ratio.(n1,n2) = n by A1,A4,BINOP_1:def 1;
            thus REAL_ratio.(frequency,f2) = s9 / r9 by A2,A7,A6,BINOP_1:def 1
                                          .= n by A2,XCMPLX_1:89;
          end;
          hence REAL_ratio.(n1,n2) = REAL_ratio.(frequency,f2);
        end;
        then n1,n2 equiv frequency,f2 by Def08a;
        hence [frequency,f2] in Class(the Equidistance of MS,[1,n])
          by EQREL_1:18;
      end;
      hence n-harmonique(MS,frequency) = n * fr by Def08b;
    end;
    hence thesis;
  end;
