Cancellation Laws/Corollary 2/Proof 2

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Corollary to Cancellation Laws

$h g = g \implies h = e$


Proof

\(\displaystyle h g\) \(=\) \(\displaystyle g\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {h g} g^{-1}\) \(=\) \(\displaystyle g g^{-1}\) Group Axiom $\text G 2$: Existence of Identity Element
\(\displaystyle \leadsto \ \ \) \(\displaystyle h \paren {g g^{-1} }\) \(=\) \(\displaystyle g g^{-1}\) Group Axiom $\text G 1$: Associativity
\(\displaystyle \leadsto \ \ \) \(\displaystyle h e\) \(=\) \(\displaystyle e\) Group Axiom $\text G 3$: Existence of Inverse Element
\(\displaystyle \leadsto \ \ \) \(\displaystyle h\) \(=\) \(\displaystyle e\) Group Axiom $\text G 2$: Existence of Identity Element

$\blacksquare$


Sources