let M1, M2 be ManySortedSet of [: the carrier of C, the carrier of C:]; :: thesis: ( ( for i, j being Object of C holds M1 . (i,j) = Hom (i,j) ) & ( for i, j being Object of C holds M2 . (i,j) = Hom (i,j) ) implies M1 = M2 )
assume that
A1: for i, j being Object of C holds M1 . (i,j) = Hom (i,j) and
A2: for i, j being Object of C holds M2 . (i,j) = Hom (i,j) ; :: thesis: M1 = M2

now :: thesis: for i, j being Object of C holds M1 . (i,j) = M2 . (i,j)

let i, j be Object of C; :: thesis: M1 . (i,j) = M2 . (i,j)
thus M1 . (i,j) = Hom (i,j) by A1
.= M2 . (i,j) by A2 ; :: thesis: verum

end;

hence M1 = M2 by ALTCAT_1:7; :: thesis: verum