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

theorem Th66:
  |.z".| = |.z.|"
proof
  per cases;
  suppose
A1: z <> 0;
    set r2i2 = (Re z)^2+(Im z)^2;
A2: r2i2 <> 0 by A1,Th5;
A3: 0 <= r2i2 by Lm1;
    thus |.z".| = sqrt ((Re z / r2i2)^2 + (Im(z"))^2) by Th20
      .= sqrt ((Re z / r2i2)^2 + ((-Im z) / r2i2)^2) by Th20
      .= sqrt ((Re z)^2 / r2i2^2 + ((-Im z) / r2i2)^2) by XCMPLX_1:76
      .= sqrt ((Re z)^2 / r2i2^2 + (-Im z)^2 / r2i2^2) by XCMPLX_1:76
      .= sqrt ((1*r2i2) / (r2i2*r2i2)) by XCMPLX_1:62
      .= sqrt (1 / r2i2) by A2,XCMPLX_1:91
      .= 1 / |.z.| by A3,SQUARE_1:18,30
      .= |.z.|" by XCMPLX_1:215;
  end;
  suppose
A4: z = 0;
    hence |.z".| = 0" .= |.z.|" by A4;
  end;
end;
