let O be set ; :: thesis: for G being GroupWithOperators of O
for H, K, H9, K9 being strict StableSubgroup of G
for JH being normal StableSubgroup of H9 "\/" (H /\ K)
for HK being normal StableSubgroup of H /\ K st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9 "\/" (H /\ K9) & HK = (H9 /\ K) "\/" (K9 /\ H) holds
(H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic
let G be GroupWithOperators of O; :: thesis: for H, K, H9, K9 being strict StableSubgroup of G
for JH being normal StableSubgroup of H9 "\/" (H /\ K)
for HK being normal StableSubgroup of H /\ K st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9 "\/" (H /\ K9) & HK = (H9 /\ K) "\/" (K9 /\ H) holds
(H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic
let H, K, H9, K9 be strict StableSubgroup of G; :: thesis: for JH being normal StableSubgroup of H9 "\/" (H /\ K)
for HK being normal StableSubgroup of H /\ K st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9 "\/" (H /\ K9) & HK = (H9 /\ K) "\/" (K9 /\ H) holds
(H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic
reconsider GG = H as GroupWithOperators of O ;
set G9 = H /\ K;
set L = H /\ K9;
reconsider G9 = H /\ K as strict StableSubgroup of GG by Lm33;
let JH be normal StableSubgroup of H9 "\/" (H /\ K); :: thesis: for HK being normal StableSubgroup of H /\ K st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9 "\/" (H /\ K9) & HK = (H9 /\ K) "\/" (K9 /\ H) holds
(H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic
let HK be normal StableSubgroup of H /\ K; :: thesis: ( H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9 "\/" (H /\ K9) & HK = (H9 /\ K) "\/" (K9 /\ H) implies (H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic )
assume that
A1: H9 is normal StableSubgroup of H and
A2: K9 is normal StableSubgroup of K ; :: thesis: ( not JH = H9 "\/" (H /\ K9) or not HK = (H9 /\ K) "\/" (K9 /\ H) or (H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic )
A3: H /\ K9 is normal StableSubgroup of G9 by A2, Th60;
reconsider N9 = H9 as normal StableSubgroup of GG by A1;
assume that
A4: JH = H9 "\/" (H /\ K9) and
A5: HK = (H9 /\ K) "\/" (K9 /\ H) ; :: thesis: (H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic
reconsider N = N9 as StableSubgroup of GG ;
set N1 = G9 /\ N;
A6: G9 "\/" N = (H /\ K) "\/" H9 by Th86
.= H9 "\/" (H /\ K) ;
reconsider L = H /\ K9 as StableSubgroup of GG by A3, Th11;
G9 /\ N = (H /\ K) /\ H9 by Th39;
then A7: L "\/" (G9 /\ N) = (H /\ K9) "\/" ((H /\ K) /\ H9) by Th86
.= ((H9 /\ H) /\ K) "\/" (K9 /\ H) by Th20
.= HK by A1, A5, Lm21 ;
reconsider HH = GG ./. N9 as GroupWithOperators of O ;
reconsider f = nat_hom N9 as Homomorphism of GG,HH ;
A8: N = Ker f by Th48;
L "\/" N = (H /\ K9) "\/" H9 by Th86
.= JH by A4 ;
hence (H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic by A3, A7, A8, A6, Th90; :: thesis: verum