
theorem Th11:
  for X be non empty set, f be PartFunc of X,ExtREAL holds (for x
  be set st x in dom f holds f.x < +infty) iff f is without+infty
proof
  let X be non empty set, f be PartFunc of X,ExtREAL;
  hereby
    assume
A1: for x be set st x in dom f holds f.x < +infty;
    now
      let x be object;
      per cases;
      suppose
        x in dom f;
        hence f.x < +infty by A1;
      end;
      suppose
        not x in dom f;
        hence f.x < +infty by FUNCT_1:def 2;
      end;
    end;
    hence f is without+infty;
  end;
  assume f is without+infty;
  hence thesis;
end;
