(Group Theory)

Given a homomorphism H between two groups G1 and G2, the homomorphism's "kernel" (symbolized Ker H) is the set of elements of G1 that give rise to the identity element of G2.

If we call the identity element of G2 "i2", we can say

g e Ker H -> H(g) = i2

Ker H necessarily contains the identity element i1 of G1.

Since H is a homomorphism, for all p, q e G1,

G2(H(p), H(q)) = H (G1(p, q)).

Let a = H (i1).  Now, for any g e G1,

G2(H(i1), H(g)) = H (G1(i1, g))

But this means that

G2(a, H(g)) = H (g)

for all g e G1,  Although Im H is not necessarily G2, G2 is still a group. Therefore, a = i2, meaning i1 e Ker H, which was to be proven.