theorem
  union Y c= Z & X in Y implies X c= Z
proof
  assume that
A1: union Y c= Z and
A2: X in Y;
  thus X c= Z
  proof
    let x be object;
    assume x in X;
    then x in union Y by A2,TARSKI:def 4;
    hence thesis by A1;
  end;
end;
