
theorem Th10:
  for X be non empty set, f be PartFunc of X,ExtREAL holds (for x
  be set st x in dom f holds -infty < f.x) 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 -infty < f.x;
    now
      let x be object;
      per cases;
      suppose
        x in dom f;
        hence -infty < f.x by A1;
      end;
      suppose
        not x in dom f;
        hence -infty < f.x by FUNCT_1:def 2;
      end;
    end;
    hence f is without-infty;
  end;
  assume f is without-infty;
  hence thesis;
end;
