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

theorem
  |.x+y.| = |.x.| + |.y.| implies 0 <= x * y
proof
A1: x * y < 0 implies x < 0 & 0 < y or 0 < x & y < 0
  proof
    assume
A2: x * y < 0;
    then x <> 0 & y <> 0;
    hence thesis by A2;
  end;
A3: x < 0 & 0 < y & x + y < 0 implies |.x+y.| <> |.x.| + |.y.|
  proof
    assume that
A4: x < 0 and
A5: 0 < y and
A6: x + y < 0;
    -(1 * x) + -y < -x + y by A5,XREAL_1:6;
    then
A7: -( 1 * ( x + y ) ) < -x + y;
    |.x.| = -x & |.y.| = y by A4,A5,Def1;
    hence thesis by A6,A7,Def1;
  end;
A8: 0 < x & y < 0 & x + y < 0 implies |.x+y.| <> |.x.| + |.y.|
  proof
    assume that
A9: 0 < x and
A10: y < 0 and
A11: x + y < 0;
    -(1 * x) + -y < x + -y by A9,XREAL_1:6;
    then
A12: - (1 * ( x + y )) < x + -y;
    |.x.| = x & |.y.| = -y by A9,A10,Def1;
    hence thesis by A11,A12,Def1;
  end;
A13: 0 < x & y < 0 & 0 <= x + y implies |.x+y.| <> |.x.| + |.y.|
  proof
    assume that
A14: 0 < x and
A15: y < 0 and
A16: 0 <= x + y;
A17: |.y.| = -y by A15,Def1;
    x + y < x + -y & |.x.| = x by A14,A15,Def1,XREAL_1:6;
    hence thesis by A16,A17,Def1;
  end;
A18: x < 0 & 0 < y & 0 <= x + y implies |.x+y.| <> |.x.| + |.y.|
  proof
    assume that
A19: x < 0 and
A20: 0 < y and
A21: 0 <= x + y;
A22: |.y.| = y by A20,Def1;
    x + y < -x + y & |.x.| = -x by A19,Def1,XREAL_1:6;
    hence thesis by A21,A22,Def1;
  end;
  assume |.x+y.| = |.x.| + |.y.| & 0 > x * y;
  hence contradiction by A1,A3,A18,A8,A13;
end;
