(1) means for given h ∈ H {\displaystyle {}h\in H} that we can write x h = h ~ x {\displaystyle {}xh={\tilde {h}}x} with some h ~ ∈ H {\displaystyle {}{\tilde {h}}\in H} . Multiplication by x − 1 {\displaystyle {}x^{-1}} from the right yields x h x − 1 = h ~ ∈ H {\displaystyle {}xhx^{-1}={\tilde {h}}\in H} ; therefore, ( 2 ) {\displaystyle {}(2)} holds. Reading this argument backwards gives the implication ( 2 ) ⇒ ( 1 ) {\displaystyle {}(2)\Rightarrow (1)} . Moreover, ( 2 ) {\displaystyle {}(2)} is an explicit reformulation of ( 3 ) {\displaystyle {}(3)} .
that we can write
with some
.
Multiplication by 
from the right yields
;
therefore, 
holds. Reading this argument backwards gives the implication 
. Moreover, 
.