A1: now

assume {} in rng (Carrier J) ; :: thesis: contradiction
then consider i being set such that
A2: i in dom (Carrier J) and
A3: {} = (Carrier J) . i by FUNCT_1:def 5;
A4: dom (Carrier J) = I by PARTFUN1:def 4;
then consider R being 1-sorted such that
A5: R = J . i and
A6: {} = the carrier of R by A2, A3, PRALG_1:def 13;
dom J = I by PARTFUN1:def 4;
then R in rng J by A2, A4, A5, FUNCT_1:def 5;
then reconsider R = R as non empty 1-sorted by WAYBEL_3:def 7;
the carrier of R = {} by A6;
hence contradiction ; :: thesis: verum

end;

the carrier of (product J) = product (Carrier J) by Def3;
then the carrier of (product J) <> {} by A1, CARD_3:37;
hence not product J is empty ; :: thesis: verum