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

theorem
  sgn (x + y) <= sgn x + sgn y + 1
proof
A1: y = 0 implies sgn (x + y) <= sgn x + sgn y + 1
  proof
    assume
A2: y = 0;
    then sgn x + sgn y + 1 = sgn x + 0 + 1 by Def2
      .= sgn x + 1;
    hence thesis by A2,XREAL_1:29;
  end;
A3: 0 < x & 0 < y implies sgn (x + y) <= sgn x + sgn y + 1
  proof
    sgn x < sgn x + 1 by XREAL_1:29;
    then
A4: sgn x + 0 < sgn x + 1 + 1 by XREAL_1:8;
    assume
A5: 0 < x & 0 < y;
    then sgn x = 1 & sgn y = 1 by Def2;
    hence thesis by A5,A4,Def2;
  end;
A6: x < 0 & 0 < y implies sgn (x + y) <= sgn x + sgn y + 1
  proof
    assume that
A7: x < 0 and
A8: 0 < y;
    sgn x = -1 by A7,Def2;
    then
A9: sgn x + sgn y + 1 = 1 by A8,Def2;
    x + y < 0 or x + y = 0 or 0 < x + y;
    hence thesis by A9,Def2;
  end;
A10: 0 < x & y < 0 implies sgn (x + y) <= sgn x + sgn y +1
  proof
    assume that
A11: 0 < x and
A12: y < 0;
    sgn x = 1 by A11,Def2;
    then
A13: sgn x + sgn y + 1 = 1 + -1 + 1 by A12,Def2
      .= 1;
    x + y < 0 or x + y = 0 or 0 < x + y;
    hence thesis by A13,Def2;
  end;
A14: x < 0 & y < 0 implies sgn (x + y) <= sgn x + sgn y + 1
  proof
    assume that
A15: x < 0 and
A16: y < 0;
    sgn y = -1 by A16,Def2;
    then sgn x + sgn y + 1 = -1 by A15,Def2;
    hence thesis by A15,A16,Def2;
  end;
  x = 0 implies sgn (x + y) <= sgn x + sgn y + 1
  proof
    assume
A17: x = 0;
    then sgn x + sgn y + 1 = 0 + sgn y + 1 by Def2
      .= sgn y + 1;
    hence thesis by A17,XREAL_1:29;
  end;
  hence thesis by A3,A10,A6,A14,A1;
end;
