reserve x, y, z, r, s, t for Real;

theorem
  0 <= x * y implies |.x+y.| = |.x.| + |.y.|
proof
  assume
A1: 0 <= x * y;
  per cases by A1;
  suppose
 x * y = 0;
    then x = 0 or y = 0 by XCMPLX_1:6;
    hence thesis;
  end;
  suppose
A5: 0 < x * y;
    then
A6: x <> 0 & y <> 0;
    per cases by A5,A6;
    suppose that
A7:   0 < x & 0 < y;
      |.x.| = x & |.y.| = y by A7,Def1;
      hence thesis by A7,Def1;
    end;
    suppose that
A8:   x < 0 and
A9:   y < 0;
      |.x.| = -x by A8,Def1;
      then |.x.| + |.y.| = (-1) * x + -(1 * y) by A9,Def1
        .= - ( x + y );
      hence thesis by A8,A9,Def1;
    end;
  end;
end;
