reserve X for set;
reserve UN for Universe;

theorem Th93:
  for X being set holds rng sequence_univers X c= union_sequence_univers X
  proof
    let X be set;
    now
      let x be object;
      assume x in rng sequence_univers X;
      then consider y be object such that
A1:   y in dom sequence_univers X and
A2:   x = (sequence_univers X).y by FUNCT_1:def 3;
      reconsider x9 = x as set by TARSKI:1;
      y in NAT by A1,Def9;
      then reconsider y as Nat;
      now
        (sequence_univers X).(y+1)
          = GrothendieckUniverse x9 by A2,Def9;
        hence x9 in (sequence_univers X).(y+1) by CLASSES3:def 4;
        y + 1 in NAT;
        then y + 1 in dom sequence_univers X by Def9;
        hence (sequence_univers X).(y+1) in rng sequence_univers X
          by FUNCT_1:3;
      end;
      hence x in union_sequence_univers X by TARSKI:def 4;
    end;
    hence thesis;
  end;
