let T1, T2 be strict TopSpace; :: thesis: ( the carrier of T1 = product (Carrier J) & product_prebasis J is prebasis of T1 & the carrier of T2 = product (Carrier J) & product_prebasis J is prebasis of T2 implies T1 = T2 )
assume that
A7: the carrier of T1 = product (Carrier J) and
A8: product_prebasis J is prebasis of T1 and
A9: the carrier of T2 = product (Carrier J) and
A10: product_prebasis J is prebasis of T2 ; :: thesis: T1 = T2

now

assume {} in rng (Carrier J) ; :: thesis: contradiction
then consider i being set such that
A11: i in dom (Carrier J) and
A12: {} = (Carrier J) . i by FUNCT_1:def 3;
A13: dom (Carrier J) = I by PARTFUN1:def 2;
then consider R being 1-sorted such that
A14: R = J . i and
A15: {} = the carrier of R by A11, A12, PRALG_1:def 13;
dom J = I by PARTFUN1:def 2;
then R in rng J by A11, A13, A14, FUNCT_1:def 3;
then reconsider R = R as non empty 1-sorted by WAYBEL_3:def 7;
the carrier of R = {} by A15;
hence contradiction ; :: thesis: verum

end;

then product (Carrier J) <> {} by CARD_3:26;
then reconsider t1 = T1, t2 = T2 as non empty TopSpace by A7, A9;
t1 = t2 by A7, A8, A9, A10, CANTOR_1:17;
hence T1 = T2 ; :: thesis: verum