
theorem Th1:
for z being Complex, r be Real holds Re(r * z) = r* Re (z)
      & Im(r * z) = r* Im (z)
proof
let z be Complex, r be Real;
A2: Re(r) = r by COMPLEX1:def 1;
hence Re(r * z) = r * Re(z) - Im(r) * Im(z) by COMPLEX1:9
              .= r * Re(z) - 0 * Im(z) by COMPLEX1:def 2
              .= r * Re(z);
thus Im(r * z) = r * Im(z) + Re(z) * Im(r) by A2,COMPLEX1:9
              .= r * Im(z) + 0 * Re(z) by COMPLEX1:def 2
              .= r * Im(z);
end;
