reserve x,y,z,w for ExtReal,
  r for Real;
reserve f,g for ExtReal;

theorem Th36:
  x <= y & z <= w implies x + z <= y + w
proof
  assume that
A1: x <= y and
A2: z <= w;
  per cases;
  suppose
A3: x = +infty & z = -infty;
A4: w <> -infty implies +infty + w = +infty by Def2;
    y = +infty by A1,A3,XXREAL_0:4;
    hence thesis by A3,A4;
  end;
  suppose
A5: x = -infty & z = +infty;
A6: y <> -infty implies +infty + y = +infty by Def2;
    w = +infty by A2,A5,XXREAL_0:4;
    hence thesis by A5,A6;
  end;
  suppose
A7: y = +infty & w = -infty;
A8: x <> +infty implies -infty + x = -infty by Def2;
    z = -infty by A2,A7,XXREAL_0:6;
    hence thesis by A7,A8;
  end;
  suppose
A9: y = -infty & w = +infty;
A10: z <> +infty implies -infty + z = -infty by Def2;
    x = -infty by A1,A9,XXREAL_0:6;
    hence thesis by A9,A10;
  end;
  suppose
A11: not ( x = +infty & z = -infty or x = -infty & z = +infty ) & not
    ( y = +infty & w = -infty or y = -infty & w = +infty );
    reconsider a = x + z, b = y + w as Element of ExtREAL by XXREAL_0:def 1;
A12: not(a = +infty & b = -infty)
    proof
      assume that
A13:  a = +infty and
A14:  b = -infty;
      x = +infty or z = +infty by A13,Lm8;
      hence thesis by A1,A2,A11,A14,Lm9,XXREAL_0:4;
    end;
A15: a in REAL & b in REAL implies a <= b
    proof
      assume
A16:  a in REAL & b in REAL;
      then
A17:  z in REAL & w in REAL by A11,Th20;
      x in REAL & y in REAL by A11,A16,Th20;
      then consider Ox,Oy,Os,Ot being Real such that
A18:  Ox = x & Oy = y & Os = z & Ot = w and
A19:  Ox <= Oy & Os <= Ot by A1,A2,A17;
      Ox + Os <= Os + Oy & Os + Oy <= Ot + Oy by A19,XREAL_1:6;
      hence thesis by A18,XXREAL_0:2;
    end;
A20: a = +infty & b in REAL implies a <= b
    proof
      assume that
A21:  a = +infty and
      b in REAL;
      x = +infty or z = +infty by A21,Lm8;
      then y = +infty or w = +infty by A1,A2,XXREAL_0:4;
      then b = +infty by A11,Def2;
      hence thesis by XXREAL_0:4;
    end;
A22: a in REAL & b = -infty implies a <= b
    proof
      assume that
      a in REAL and
A23:  b = -infty;
      y = -infty or w = -infty by A23,Lm9;
      then x = -infty or z = -infty by A1,A2,XXREAL_0:6;
      then a = -infty by A11,Def2;
      hence thesis by XXREAL_0:5;
    end;
    a in REAL & b in REAL or a in REAL & b = +infty or a in REAL & b =
-infty or a = +infty & b in REAL or a = +infty & b = +infty or a = +infty & b =
-infty or a = -infty & b in REAL or a = -infty & b = +infty or a = -infty & b =
    -infty by XXREAL_0:14;
    hence thesis by A15,A22,A20,A12,XXREAL_0:4,5;
  end;
end;
