let I be non empty finite set ; :: thesis: for F being Group-like associative multMagma-Family of I
for x being set st x in the carrier of (product F) holds
x is I -defined total Function
let F be Group-like associative multMagma-Family of I; :: thesis: for x being set st x in the carrier of (product F) holds
x is I -defined total Function
let x be set ; :: thesis: ( x in the carrier of (product F) implies x is I -defined total Function )
assume A1: x in the carrier of (product F) ; :: thesis: x is I -defined total Function
D1: dom (Carrier F) = I by PARTFUN1:def 2;
the carrier of (product F) = product (Carrier F) by GROUP_7:def 2;
then consider f being Function such that
D2: ( x = f & dom f = dom (Carrier F) & ( for y being object st y in dom (Carrier F) holds
f . y in (Carrier F) . y ) ) by CARD_3:def 5, A1;
thus x is I -defined total Function by D2, D1, RELAT_1:def 18, PARTFUN1:def 2; :: thesis: verum