
theorem Th4:
  for X being non empty set holds X is axiom_GU4 iff X is FamUnion-closed
  proof
    let X be non empty set;
    hereby
      assume
A1:   X is axiom_GU4;
      now
        let A be set;
        let f be Function;
        assume that
A2:     dom f = A and
A3:     rng f c= X and
A4:     A in X;
        reconsider f9 = f as Function of A,X by A2,A3,FUNCT_2:2;
        reconsider f9 as X-valued ManySortedSet of A;
        union rng f9 in X by A4,A1;
        hence union rng f in X;
      end;
      hence X is FamUnion-closed;
    end;
    assume
A5: X is FamUnion-closed;
    now
      let I be Element of X;
      let x be X-valued ManySortedSet of I;
      dom x = I by PARTFUN1:def 2;
      hence union rng x in X by A5;
    end;
    hence X is axiom_GU4;
  end;
