
theorem Th11:
  for f1,f2,fn1,fm1,fn2,fm2 being Element of REAL_Music
  for r1,r2 being positive Real :::st f1 = r1 & f2 = r2
    holds
  for n,m being non zero Nat st fn1 = n * r1 & fm1 = m * r1 &
  fn2 = n * r2 & fm2 = m * r2 holds fn1,fm1 equiv fn2,fm2
  proof
    let f1,f2,fn1,fm1,fn2,fm2 be Element of REAL_Music;
    let r1,r2 be positive Real;
:::    assume f1 = r1 & f2 = r2;
    now
      let n,m be non zero Nat;
      assume
A0:   fn1 = n * r1 & fm1 = m * r1 & fn2 = n * r2 & fm2 = m * r2;
      reconsider z = [fn1,fm1] as Element of [:REALPLUS,REALPLUS:]
        by ZFMISC_1:def 2;
      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;
      reconsider z9 = [fn2,fm2] as Element of [:REALPLUS,REALPLUS:]
        by ZFMISC_1:def 2;
      consider x99,y99 be Element of REALPLUS such that
A4:   z9 = [x99,y99] and
A5:   REAL_ratio.z9 = 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;
      now
        thus REAL_ratio.(fn1,fm1) = s / r by A3,A2,BINOP_1:def 1;
        thus REAL_ratio.(fn2,fm2) = s9 / r9 by A6,A5,BINOP_1:def 1;
        r = n * r1 & s = m * r1 &
          r9 = n * r2 & s9 = m * r2 by A4,A6,A1,A3,A0,XTUPLE_0:1;
        then s / r = (m qua Real) / (n qua Real) &
          s9 / r9 = (m qua Real) / (n qua Real) by XCMPLX_1:91;
        hence s / r = s9 / r9;
      end;
      hence fn1,fm1 equiv fn2,fm2 by Th7;
    end;
    hence thesis;
  end;
