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

theorem Th23:
  Re z = 0 & Im z <> 0 implies Re(z") = 0 & Im(z") = -(Im z)"
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") = 0 by A1;
  Im(z") = (- Im z) / ((Re z)^2+(Im z)^2) by Th20;
  hence Im(z") = -(1*Im z / (Im z * Im z)) by A1,XCMPLX_1:187
    .= -(1 / Im z) by A2,XCMPLX_1:91
    .= - (Im z)" by XCMPLX_1:215;
end;
