Theorem-G satises the right and left cancellation laws
Let G be a group.
(i) G satises the right and left cancellation laws; that is, if a; b; x ≡ G, then ax = bx and xa = xb each imply that a = b.
(ii) If g ≡ G, then (g^{-1})^{-1} = g.
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Proof:
(i) From ax = bx, we have axx^{-1} = bxx^{-1}, then ae = be, then a = b. Similarly for the other case.(ii) Temporarily denote the inverse of g^{-1} by h (instead of (g^{-1})^{-1}). Then the defining property of h, from the axiom for inverses applied to g^{-1}, is that
g^{-1}h = hg^{-1} = e:
But g itself satises these equations in place of h, because the axiom for inverses applied to g says that
gg^{-1} = g^{-1}g = e:
Hence, since inverses are unique, h = (g^{-1})^{-1} = g, as required.
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