reserve x, y for object, I for set,
  A, B, X, Y for ManySortedSet of I;

theorem :: Tarski:def 4
  A in union X iff ex Y st A in Y & Y in X
proof
  thus A in union X implies ex Y st A in Y & Y in X
  proof
    defpred P[object,object] means
      ex B being set st B = $2 & A.$1 in B & $2 in X.$1;
    assume
A1: A in union X;
A2: for i being object st i in I ex Y being object st P[i,Y]
    proof
      let i be object;
      assume
A3:   i in I;
      then A.i in (union X).i by A1;
      then A.i in union (X.i) by A3,Def2;
      then consider B being set such that
A4:     A.i in B & B in X.i by TARSKI:def 4;
      take B;
      thus thesis by A4;
    end;
    consider K be ManySortedSet of I such that
A5: for i be object st i in I holds P[i,K.i] from PBOOLE:sch 3(A2);
    take K;
    thus A in K
    proof
      let i be object;
      assume i in I;
       then P[i,K.i] by A5;
      hence thesis;
    end;
    thus K in X
    proof
      let i be object;
      assume i in I;
       then P[i,K.i] by A5;
      hence thesis;
    end;
  end;
  given Y such that
A6: A in Y & Y in X;
  let i be object;
  assume
A7: i in I;
  then A.i in Y.i & Y.i in X.i by A6;
  then A.i in union (X.i) by TARSKI:def 4;
  hence thesis by A7,Def2;
end;
