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

theorem
  not ( x = +infty & y = +infty or x = -infty & y = -infty or y = +infty
  & z = -infty or y = -infty & z = +infty ) & x - y <= z implies y <> -infty
proof
  assume that
A1: not ( x=+infty & y=+infty or x=-infty & y=-infty or y = +infty & z =
  -infty or y = -infty & z = +infty ) and
A2: x - y <= z;
    assume
A3: y = -infty;
    then x - y = +infty by A1,Th14;
    hence contradiction by A1,A2,A3,XXREAL_0:4;
end;
