# Theorem

Let H be subgroup of Group G. Let ${\displaystyle \ast }$ be the binary operation of both H and G

H and G shares identity

# Proof

 0. Let eH, eG be identities of H and G respectively. 1. ${\displaystyle {\color {OliveGreen}e_{H}}\ast {\color {OliveGreen}e_{H}}={\color {OliveGreen}e_{H}}}$ eH is identity of H (usage 1, 3) 2. ${\displaystyle {\color {OliveGreen}e_{H}}\in H}$ eH is identity of H (usage 1) 3. ${\displaystyle H\subseteq G}$ H is subgroup of G 4. ${\displaystyle {\color {OliveGreen}e_{H}}\in G}$ 2. and 3. 5. ${\displaystyle {\color {OliveGreen}e_{H}}\ast {\color {Blue}e_{G}}={\color {OliveGreen}e_{H}}}$ 4. and eG is identity of G (usage 3) 6. ${\displaystyle {\color {OliveGreen}e_{H}}\ast {\color {Blue}e_{G}}={\color {OliveGreen}e_{H}}\ast {\color {OliveGreen}e_{H}}}$ 1. and 5. 7. ${\displaystyle {\color {Blue}e_{G}}={\color {OliveGreen}e_{H}}}$ cancellation on group G

# Usages

1. If H is subgroup of group G, identity of G is identity of H.
2. If H is subgroup of group G, identity of G is in H.