
theorem Th7:
  for X being non empty set holds union SmallestPartition X = X
  proof
    let X be non empty set;
A0: SmallestPartition X =
      the set of all {x} where x is Element of X by EQREL_1:37;
    hereby
      let o be object;
      assume o in union SmallestPartition X;
      then ex y be set st o in y in SmallestPartition X by TARSKI:def 4;
      hence o in X;
    end;
    let o be object;
    assume o in X;
    then o in {o} in SmallestPartition X by A0,TARSKI:def 1;
    hence thesis by TARSKI:def 4;
  end;
