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

theorem
  x * y <> +infty & x * y <> -infty implies x in REAL or y in REAL
proof
  assume that
A1: x * y <> +infty and
A2: x * y <> -infty;
  assume
A3: ( not x in REAL)& not y in REAL;
  per cases by A3,XXREAL_0:14;
  suppose
    x = +infty & y = +infty;
    hence contradiction by A1,Def5;
  end;
  suppose
    x = +infty & y = -infty;
    hence contradiction by A2,Def5;
  end;
  suppose
    x = -infty & y = +infty;
    hence contradiction by A2,Def5;
  end;
  suppose
    x = -infty & y = -infty;
    hence contradiction by A1,Def5;
  end;
end;
