Normal Subgroup of Product of Groups

Let I be a non empty set, let F be a group-like multiplicative magma family of I, and let i be an element of I. Note that F (i) is group-like. Let I be a non empty set, let F be an associative multiplicative magma family of I, and let i be an element of I. Observe that F (i) is associative. Let I be a non empty set, let F be a commutative multiplicative magma family of I, and let i be an element of I. Note that F (i) is commutative. In the sequel I is a non empty set, F is an associative group-like multiplicative magma family of I, and i, j are elements of I. We now state the proposition (1) Let x be a function and g be an element of F (i). Then domx = I and x(i) = g and for every element j of I such that j 6= i holds x(j) = 1F (j) if and only if x = 1∏F +· (i, g). Let I be a non empty set, let F be an associative group-like multiplicative magma family of I, and let i be an element of I. The functor ProjSet(F, i) yields a subset of ∏ F and is defined by: