deffunc H₁( Element of dom (carr G)) -> Element of the carrier of (G . $1) = the ZeroF of (G . $1);
consider p being non empty FinSequence such that
A13: ( len p = len (carr G) & ( for j being Element of dom (carr G) holds p . j = H₁(j) ) ) from PRVECT_1:sch 1();
A14: dom (carr G) = Seg (len (carr G)) by FINSEQ_1:def 3;
A15: dom G = Seg (len G) by FINSEQ_1:def 3;

A16: now :: thesis: for i being object st i in dom (carr G) holds
p . i in (carr G) . i

let i be object ; :: thesis: ( i in dom (carr G) implies p . i in (carr G) . i )
assume i in dom (carr G) ; :: thesis: p . i in (carr G) . i
then reconsider x = i as Element of dom (carr G) ;
reconsider x9 = x as Element of dom G by A15, A14, Def4;
( p . x = the ZeroF of (G . x) & (carr G) . x9 = the carrier of (G . x9) ) by A13, Def4;
hence p . i in (carr G) . i ; :: thesis: verum

end;

dom p = Seg (len p) by FINSEQ_1:def 3;
then reconsider t = p as Element of product (carr G) by A13, A14, A16, CARD_3:9;
take t ; :: thesis: for j being Element of dom (carr G) holds t . j = the ZeroF of (G . j)
thus for j being Element of dom (carr G) holds t . j = the ZeroF of (G . j) by A13; :: thesis: verum