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

theorem
  x = |.x.| * sgn x
proof
A1: 0 < x implies x = |.x.| * sgn x
  proof
    assume
A2: 0 < x;
    then |.x.| = x by Def1;
    then |.x.| * sgn x = x * 1 by A2,Def2;
    hence thesis;
  end;
A3: x < 0 implies x = |.x.| * sgn x
  proof
    assume
A4: x < 0;
    then |.x.| = -x by Def1;
    then |.x.| * sgn x = (-x) * (-1) by A4,Def2
      .= x;
    hence thesis;
  end;
  x = 0 implies x = |.x.| * sgn x;
  hence thesis by A1,A3;
end;
