theorem
  Union the Sorts of T = union {T height<= i: not contradiction}
  proof
    thus Union the Sorts of T c= union {T height<= i: not contradiction}
    proof
      let a; assume a in Union the Sorts of T;
      then reconsider t = a as Element of T;
      t in T height<= height t &
      T height<= height t in {T height<= i: not contradiction} by Th10a;
      hence thesis by TARSKI:def 4;
    end;
    let a; assume a in union {T height<= i: not contradiction};
    then consider I such that
A1: a in I & I in {T height<= i: not contradiction} by TARSKI:def 4;
    consider i such that
A2: I = T height<= i by A1;
    thus thesis by A1,A2;
  end;
