Let x 1 , x 2 ∈ L {\displaystyle {}x_{1},x_{2}\in L} be given with f ( x 1 ) = f ( x 2 ) {\displaystyle {}f(x_{1})=f(x_{2})} . We have to show x 1 = x 2 {\displaystyle {}x_{1}=x_{2}} . We have
Since by assumption g ∘ f {\displaystyle {}g\circ f} is injective, we get x 1 = x 2 {\displaystyle {}x_{1}=x_{2}} .
