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

theorem Th22:
  Re z <> 0 & Im z = 0 implies Re(z") = (Re z)" & Im(z") = 0
proof
  assume that
A1: Re z <> 0 and
A2: Im z = 0;
  Re(z") = Re z / ((Re z)^2+(Im z)^2) by Th20;
  hence Re(z") = (1*Re z) / (Re z * Re z) by A2
    .= 1/Re z by A1,XCMPLX_1:91
    .= (Re z)" by XCMPLX_1:215;
  Im(z") = (- Im z) / ((Re z)^2+(Im z)^2) by Th20;
  hence thesis by A2;
end;
