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

theorem Th49:
  for frequency be Element of RAT_Music holds
  ex fr,qr being positive Rational st fr = frequency &
  qr = (1 qua Rational) / 2 * fr & [qr,fr] in octave(RAT_Music)
  proof
    set MS = RAT_Music;
    now
      let frequency be Element of MS;
      reconsider f = frequency as positive Rational by Th2;
      reconsider qr = (1 qua Rational) / 2 * f as positive Rational;
      reconsider q = qr as Element of MS by Th2;
      take f,qr;
      thus f = frequency;
      thus qr = (1 qua Rational) / 2 * f;
      reconsider n2 = 2,n3 = 1 as Element of MS by Th20;
      reconsider x = [n2,n3],y = [frequency,q] as
        Element of [:RATPLUS,RATPLUS:];
      reconsider z = [frequency,q] as Element of [:RATPLUS,RATPLUS:];
      consider x9,y9 be Element of RATPLUS such that
A1:   z = [x9,y9] and
A2:   RAT_ratio.z = RAT_ratio(x9,y9) by Def05;
      consider r,s be positive Rational such that
A3:   x9 = r & s = y9 & RAT_ratio(x9,y9) = s / r by Def04;
      consider x99,y99 be Element of RATPLUS such that
A4:   x = [x99,y99] and
A5:   RAT_ratio.x = RAT_ratio(x99,y99) by Def05;
      consider r9,s9 be positive Rational such that
A6:   x99 = r9 & s9 = y99 & RAT_ratio(x99,y99) = s9 / r9 by Def04;
A7:   n2 = r9 & n3 = s9 & r = frequency & s = q
        by A3,A1,A4,A6,XTUPLE_0:1;
      now
        thus RAT_ratio.(n2,n3) = (1 qua Real) / (2 qua Real)
          by A6,A7,A5,BINOP_1:def 1;
        thus RAT_ratio.(frequency,q) = s / r by A2,A3,BINOP_1:def 1
                                    .= (1 qua Rational) / 2 by A7,XCMPLX_1:89;
      end;
      then n2,n3 equiv frequency,q by Def08a;
      then n3,n2 equiv q,frequency by Th28;
      hence [qr,f] in octave(MS) by EQREL_1:18;
    end;
    hence thesis;
  end;
