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

theorem
  not (x = +infty & y = -infty) & not (x = -infty & y = +infty) & not (y
  = +infty & z = +infty) & x <= z - y implies y <> +infty & x + y <= z
proof
  assume that
A1: not (x = +infty & y = -infty) and
A2: not (x = -infty & y = +infty) and
A3: not (y = +infty & z = +infty) and
A4: x <= z - y;
  thus
A5: y <> +infty
  proof
    assume
A6: y = +infty;
    then z - y = -infty by A3,Th13;
    hence contradiction by A2,A4,A6,XXREAL_0:6;
  end;
  per cases by A5;
  suppose
    y = -infty;
    then x + y = -infty by A1,Def2;
    hence thesis by XXREAL_0:5;
  end;
  suppose
A7: y <> +infty & y <> -infty;
    - -infty = +infty by Def3;
    then
A8: -y <> -infty by A7;
    - +infty = -infty by Def3;
    then -y <> +infty by A7;
    then z - y + y = z + (-y + y) by A7,A8,Th29
      .= z + 0 by Th7
      .= z by Th4;
    hence thesis by A4,Th36;
  end;
end;
