reserve Omega for set;
reserve m,n,k for Nat;
reserve x,y for object;
reserve r,r1,r2,r3 for Real;
reserve seq,seq1 for Real_Sequence;
reserve Sigma for SigmaField of Omega;
reserve ASeq,BSeq for SetSequence of Sigma;
reserve A, B, C, A1, A2, A3 for Event of Sigma;

theorem Th1:
  for r,r1,r2,r3 st r <> 0 & r1 <> 0 holds (r3/r1 = r2/r iff r3 * r = r2 * r1)
proof
  let r,r1,r2,r3;
  assume that
A1: r <> 0 and
A2: r1 <> 0;
  thus r3/r1 = r2/r implies r3 * r = r2 * r1
  proof
    assume
A3: r3/r1 = r2/r;
    r3 * r = r3/r1 * r1 * r by A2,XCMPLX_1:87
      .= r2/r * r * r1 by A3
      .= r2 * r1 by A1,XCMPLX_1:87;
    hence thesis;
  end;
  assume
A4: r3 * r = r2 * r1;
  r3/r1 = (r3 * 1)/r1 .= (r3 * (r * r"))/r1 by A1,XCMPLX_0:def 7
    .= (r2 * r1 * r")/r1 by A4,XCMPLX_1:4
    .= (r2 * r" * r1)/r1
    .= (r2/r * r1)/r1 by XCMPLX_0:def 9
    .= (r2/r * r1) * r1" by XCMPLX_0:def 9
    .= r2/r * (r1 * r1")
    .= r2/r * 1 by A2,XCMPLX_0:def 7
    .= r2/r;
  hence thesis;
end;
