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

theorem Th41:
  for frequency be Element of REAL_Music holds
  ex fr,qr being positive Real st fr = frequency & qr = 2 * fr &
  [fr,qr] in octave(REAL_Music)
  proof
    set MS = REAL_Music;
    now
      let frequency be Element of MS;
      reconsider f = frequency as positive Real by Th1;
      reconsider qr = 2 * f as positive Real;
      reconsider q = qr as Element of MS by Th1;
      take f,qr;
      thus f = frequency;
      thus qr = 2 * f;
      reconsider n2 = 1,n3 = 2 as Element of MS by Th20;
      reconsider x = [n2,n3],y = [frequency,q] as
        Element of [:REALPLUS,REALPLUS:];
      reconsider z = [frequency,q] as Element of [:REALPLUS,REALPLUS:];
      consider x9,y9 be Element of REALPLUS such that
A1:   z = [x9,y9] and
A2:   REAL_ratio.z = REAL_ratio(x9,y9) by Def02;
      consider r,s be positive Real such that
A3:   x9 = r & s = y9 & REAL_ratio(x9,y9) = s / r by Def01;
      consider x99,y99 be Element of REALPLUS such that
A4:   x = [x99,y99] and
A5:   REAL_ratio.x = REAL_ratio(x99,y99) by Def02;
      consider r9,s9 be positive Real such that
A6:   x99 = r9 & s9 = y99 & REAL_ratio(x99,y99) = s9 / r9 by Def01;
A7:   n2 = r9 & n3 = s9 & r = frequency & s = q
        by A3,A1,A4,A6,XTUPLE_0:1;
      now
        thus REAL_ratio.(n2,n3) = (2 qua Real) / (1 qua Real)
          by A7,A6,A5,BINOP_1:def 1;
        thus REAL_ratio.(frequency,q) = REAL_ratio.z by BINOP_1:def 1
                                     .= 2 by A2,A3,A7,XCMPLX_1:89;
      end;
      then n2,n3 equiv frequency,q by Def08a;
      hence [f,qr] in octave(MS) by EQREL_1:18;
    end;
    hence thesis;
  end;
