*Let f:G→H be a homomorphism of groups with operators, N the kernel of f, G′ a stable subgroup of G and L a normal stable subgroup of G′. Then LN,L.(G′∩N) and f(L) are normal stable subgroups of G′N,G′ and f(G′) respectively and the three quotient groups with operators G′N/LN,G′/L.(G′∩N) and f(G′)/f(L) are isomorphic.
Let H′=f(G′) and f′ denote the homomorphism of G′ onto H′ which coincides with f on G′; the kernel of f′ is G∩N and f′(L)=f(L); by Proposition 7, f′(L) is a normal stable subgroup of H′ and
f′−1(f′(L))=L.(G′∩N) is a normal stable subgroup of
G′. Let
λ be the canonical homomorphism of
H′ onto
H′/f′(L)=f(G′)/f(L): as
λ∘f′ is surjective with kernel
f′−1(f′(L))=L.(G′∩N),
it defines an isomorphism of G′/L⋅(G′∩N) onto f(G′)/f(L). By Proposition 7, (b), f−1(H′)=G′N; if f′′ is the homomorphism of G′N onto H′ which coincides with f on G′N, the homomorphism λ∘f′′ of G′N onto f(G′)/f(L) is surjective with kernel f−1(f(L))=LN; this proves that LN is a normal stable subgroup of G′N and that λ∘f′′ defines an isomorphism of G′N/LN onto f(G′)/f(L).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.