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

theorem
  sgn x + sgn y - 1 <= sgn ( x + y )
proof
A1: x = 0 or y = 0 implies sgn x + sgn y - 1 <= sgn (x + y)
  proof
A2: y = 0 implies sgn x + sgn y - 1 <= sgn (x + y)
    proof
A3:   sgn x - 1 < sgn x + -1 + 1 by XREAL_1:29;
      assume
A4:   y = 0;
      then sgn x + sgn y - 1 = sgn x + 0 - 1 by Def2
        .= sgn x - 1;
      hence thesis by A4,A3;
    end;
A5: x = 0 implies sgn x + sgn y - 1 <= sgn (x + y)
    proof
A6:   sgn y - 1 < sgn y + -1 + 1 by XREAL_1:29;
      assume
A7:   x = 0;
      then sgn x + sgn y - 1 = 0 + sgn y - 1 by Def2
        .= sgn y - 1;
      hence thesis by A7,A6;
    end;
    assume x = 0 or y = 0;
    hence thesis by A5,A2;
  end;
A8: x < 0 & y < 0 implies sgn x + sgn y - 1 <= sgn (x + y)
  proof
    assume that
A9: x < 0 and
A10: y < 0;
    sgn x = -1 by A9,Def2;
    then
A11: sgn x = sgn (x + y) by A9,A10,Def2;
A12: sgn (x + y) + -1 - 1 < sgn (x + y) + -1 - 1 + 1 & sgn (x + y) + -1 <
    sgn (x + y) + -1 + 1 by XREAL_1:29;
    sgn y = -1 by A10,Def2;
    hence thesis by A11,A12,XXREAL_0:2;
  end;
A13: 0 < x & y < 0 implies sgn x + sgn y - 1 <= sgn (x + y)
  proof
    assume that
A14: 0 < x and
A15: y < 0;
    sgn x = 1 by A14,Def2;
    then
A16: sgn x + sgn y = 1 + -1 by A15,Def2
      .= 0;
    x + y < 0 or x + y = 0 or 0 < x + y;
    hence thesis by A16,Def2;
  end;
A17: x < 0 & 0 < y implies sgn x + sgn y - 1 <= sgn (x + y)
  proof
    assume that
A18: x < 0 and
A19: 0 < y;
    sgn x = -1 by A18,Def2;
    then
A20: sgn x + sgn y = -1 + 1 by A19,Def2
      .= 0;
    x + y < 0 or x + y = 0 or 0 < x + y;
    hence thesis by A20,Def2;
  end;
  0 < x & 0 < y implies sgn x + sgn y - 1 <= sgn (x + y)
  proof
    assume that
A21: 0 < x and
A22: 0 < y;
    sgn y = 1 by A22,Def2;
    then sgn x + sgn y - 1 = 1 by A21,Def2;
    hence thesis by A21,A22,Def2;
  end;
  hence thesis by A8,A17,A13,A1;
end;
