
theorem
  for X being set, M, N being Cardinal st X is mutually-disjoint &
    M c= card X & for Y being set st Y in X holds N c= card Y
  holds M*`N c= card union X
proof
  let X be set, M,N be Cardinal;
  assume that
A1: X is mutually-disjoint and
A2: M c= card X and
A3: for Y being set st Y in X holds N c= card Y;
  now
    let x be object;
    assume x in dom id X;
    then (id X).x in X by FUNCT_1:18;
    hence N c= card ((id X).x) by A3;
  end;
  then M*`N c= Sum Card id X by A2,Th12;
  hence thesis by A1, Th14;
end;
