reserve a,b,c,d for Real;
reserve z,z1,z2 for Complex;

theorem
  Im z1 = 0 & Im z2 = 0 & Re z2 <> 0 implies Re(z1/z2) = (Re z1)/(Re z2)
  & Im(z1/z2) = 0
proof
  assume that
A1: Im z1 = 0 and
A2: Im z2 = 0 & Re z2 <> 0;
A3: z1/z2 = z1*z2" & Im(z2") = 0 by A2,Th22,XCMPLX_0:def 9;
  hence Re(z1/z2) = (Re z1)*Re(z2") by A1,Th14
    .= (Re z1)*(Re z2)" by A2,Th22
    .= (Re z1)/(Re z2) by XCMPLX_0:def 9;
  thus thesis by A1,A3,Th14;
end;
