let O be set ; :: thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G
for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 & H1 = s1 . i & H2 = s1 . (i + 1) holds
H2 is normal StableSubgroup of H1
let G be GroupWithOperators of O; :: thesis: for H1, H2 being StableSubgroup of G
for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 & H1 = s1 . i & H2 = s1 . (i + 1) holds
H2 is normal StableSubgroup of H1
let H1, H2 be StableSubgroup of G; :: thesis: for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 & H1 = s1 . i & H2 = s1 . (i + 1) holds
H2 is normal StableSubgroup of H1
let s1 be CompositionSeries of G; :: thesis: for i being Nat st 1 <= i & i <= (len s1) - 1 & H1 = s1 . i & H2 = s1 . (i + 1) holds
H2 is normal StableSubgroup of H1
let i be Nat; :: thesis: ( 1 <= i & i <= (len s1) - 1 & H1 = s1 . i & H2 = s1 . (i + 1) implies H2 is normal StableSubgroup of H1 )
assume that
A1: 1 <= i and
A2: i <= (len s1) - 1 ; :: thesis: ( not H1 = s1 . i or not H2 = s1 . (i + 1) or H2 is normal StableSubgroup of H1 )
A3: i + 1 <= ((len s1) - 1) + 1 by A2, XREAL_1:6;
A4: 0 + i <= 1 + i by XREAL_1:6;
then 1 <= i + 1 by A1, XXREAL_0:2;
then i + 1 in Seg (len s1) by A3;
then A5: i + 1 in dom s1 by FINSEQ_1:def 3;
i <= len s1 by A4, A3, XXREAL_0:2;
then i in Seg (len s1) by A1, FINSEQ_1:1;
then A6: i in dom s1 by FINSEQ_1:def 3;
assume ( H1 = s1 . i & H2 = s1 . (i + 1) ) ; :: thesis: H2 is normal StableSubgroup of H1
hence H2 is normal StableSubgroup of H1 by A5, A6, Def31; :: thesis: verum