deffunc H1( Element of dom (carr G)) -> Element of bool [:[: the carrier of (G . \$1), the carrier of (G . \$1):], the carrier of (G . \$1):] = the addF of (G . \$1);
consider p being non empty FinSequence such that
A1: ( len p = len (carr G) & ( for j being Element of dom (carr G) holds p . j = H1(j) ) ) from
now :: thesis: for j being Element of dom (carr G) holds p . j is BinOp of ((carr G) . j)
let j be Element of dom (carr G); :: thesis: p . j is BinOp of ((carr G) . j)
len G = len (carr G) by Def4;
then reconsider k = j as Element of dom G by FINSEQ_3:29;
( p . j = the addF of (G . j) & the carrier of (G . k) = (carr G) . k ) by ;
hence p . j is BinOp of ((carr G) . j) ; :: thesis: verum
end;
then reconsider p9 = p as BinOps of carr G by ;
take p9 ; :: thesis: ( len p9 = len (carr G) & ( for j being Element of dom (carr G) holds p9 . j = the addF of (G . j) ) )
thus ( len p9 = len (carr G) & ( for j being Element of dom (carr G) holds p9 . j = the addF of (G . j) ) ) by A1; :: thesis: verum