reserve U for Universe;
reserve x for Element of U;

theorem Th28:
  for X being finite Subset of U holds X in U
  proof
    let X be finite Subset of U;
    per cases;
    suppose X is empty;
      hence thesis by CLASSES2:56;
    end;
    suppose X is non empty;
      then reconsider X as non empty set;
      reconsider Y = SmallestPartition X as non empty set;
      consider p be FinSequence such that
A1:   rng p = Y and
      p is one-to-one by FINSEQ_4:58;
A2:   dom p = Seg len p by FINSEQ_1:def 3;
A0:   Y = the set of all {x} where x is Element of X
        by EQREL_1:37;
      now
        let o be object;
        assume o in Y;
        then consider x be Element of X such that
A3:     o = {x} by A0;
        x in X & X c= U;
        hence o in U by A3,Th18;
      end;
      then rng p c= U by A1;
      then union rng p in U by A2,CLASSES4:5,57;
      hence thesis by A1,Th7;
    end;
  end;
