reserve X for set;
reserve UN for Universe;

theorem
  ex f being Function of NAT,union union rng sequence_univers st
  (for i being Nat holds f.i in ComplUniverse.i)
  proof
    set g = the Choice of ComplUniverse;
A1: dom g = dom ComplUniverse by ORDERS_1:def 1
         .= NAT by FUNCT_2:def 1;
    rng g c= union union rng sequence_univers
    proof
      let x be object;
      assume x in rng g;
      then consider y be object such that
A2:   y in dom g and
A3:   x = g.y by FUNCT_1:def 3;
      reconsider y as Element of NAT by A1,A2;
A4:   ComplUniverse.y is non empty by Th109;
      y in dom ComplUniverse by A1,A2,FUNCT_2:def 1;
      then x = the Element of ComplUniverse.y by A3,ORDERS_1:def 1;
      then
A5:  x in ComplUniverse.y by A4;
A6:  UNIVERSE ((y + 1) + 1) is axiom_GU1;
A7:   UNIVERSE (y + 1) in UNIVERSE ((y + 1) + 1) by Th99;
A8:   UNIVERSE ((y + 1) + 1) c= union rng sequence_univers
      proof
        let x be object;
        assume
A9:    x in UNIVERSE ((y + 1) + 1);
        set n = (y + 1) + 1;
        (n + 1) + 1 in NAT;
        then
A10:    ((n + 1) + 1) in dom sequence_univers by Def9;
A11:   x in (sequence_univers).(n + 1) &
         (sequence_univers).(n + 1) c= (sequence_univers).((n + 1) + 1)
         by Th102,A9,Th105;
        (sequence_univers).((n + 1) + 1) in rng sequence_univers
          by A10,FUNCT_1:def 3;
        hence thesis by A11,TARSKI:def 4;
      end;
      reconsider z1 = UNIVERSE (y + 1) as Element of UNIVERSE((y + 1) + 1)
        by Th99;
      UNIVERSE y in UNIVERSE (y + 1) by Th99;
      then reconsider z2 = UNIVERSE y as Element of UNIVERSE((y + 1) + 1)
        by A6,A7;
A12:   x in z1 \ z2 by A5,Def14;
      x in union UNIVERSE ((y+1)+1) &
      union UNIVERSE ((y+1)+1) c= union union rng sequence_univers
        by A12,A8,ZFMISC_1:77,TARSKI:def 4;
      hence thesis;
    end;
    then reconsider g as Function of NAT,union union rng sequence_univers
      by A1,FUNCT_2:2;
    take g;
    for i be Nat holds g.i in ComplUniverse.i
    proof
      let i be Nat;
      dom ComplUniverse = NAT by FUNCT_2:def 1;
      then g.i = the Element of (ComplUniverse).i
        by ORDERS_1:def 1,ORDINAL1:def 12;
      then g.i is Element of (ComplUniverse).i & ComplUniverse.i is non empty
        by Th109;
      hence thesis;
    end;
    hence thesis;
  end;
