theorem
  (for X st X in A holds X c= Z) implies union A c= Z
proof
  assume
A1: for X st X in A holds X c= Z;
  let y;
  assume y in union A;
  then consider Y such that
A2: y in Y and
A3: Y in A by TARSKI:def 4;
  Y c= Z by A1,A3;
  hence thesis by A2;
end;
