# Theorem

Let G be any group with operation ${\displaystyle \ast }$ .

${\displaystyle \forall \;g\in G:[g^{-1}]^{-1}=g}$
In Group G, inverse of inverse of any element g is g.

# Proof

 0. Choose ${\displaystyle {\color {OliveGreen}g}\in G}$ 1. ${\displaystyle \exists \;{\color {BrickRed}g^{-1}}\in G:{\color {OliveGreen}g}\ast {\color {BrickRed}g^{-1}}={\color {BrickRed}g^{-1}}\ast {\color {OliveGreen}g}=e_{G}}$ definition of inverse of g in G (usage 1,3) 2. ${\displaystyle {\color {OliveGreen}g}\ast {\color {BrickRed}a}={\color {BrickRed}a}\ast {\color {OliveGreen}g}=e_{G}}$ let a = g−1 3. ${\displaystyle {\color {BrickRed}a}\ast {\color {OliveGreen}g}={\color {OliveGreen}g}\ast {\color {BrickRed}a}=e_{G}}$ 4. ${\displaystyle [{\color {BrickRed}a}]^{-1}={\color {OliveGreen}g}}$ definition of inverse of a in G (usage 2) 5. ${\displaystyle [{\color {BrickRed}g^{-1}}]^{-1}={\color {OliveGreen}g}}$ as a = g−1

# Diagrams

 1. inverse of filled circle is empty circle. 2. inverse of empty circle is filled circle, given 1.