theorem
  (card X c= M & for Y st Y in X holds card Y c= N) implies card union X c=
  M*`N
proof
  assume that
A1: card X c= M and
A2: for Y st Y in X holds card Y c= N;
  now
    let x;
    assume x in dom id X;
    then (id X).x in X by FUNCT_1:18;
    hence card ((id X).x) c= N by A2;
  end;
  then card Union id X c= M*`N by A1,Th85;
  hence thesis;
end;
