let O be set ; :: thesis: for G being GroupWithOperators of O
for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 holds
( s1 . i is strict StableSubgroup of G & s1 . (i + 1) is strict StableSubgroup of G )
let G be GroupWithOperators of O; :: thesis: for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 holds
( s1 . i is strict StableSubgroup of G & s1 . (i + 1) is strict StableSubgroup of G )
let s1 be CompositionSeries of G; :: thesis: for i being Nat st 1 <= i & i <= (len s1) - 1 holds
( s1 . i is strict StableSubgroup of G & s1 . (i + 1) is strict StableSubgroup of G )
let i be Nat; :: thesis: ( 1 <= i & i <= (len s1) - 1 implies ( s1 . i is strict StableSubgroup of G & s1 . (i + 1) is strict StableSubgroup of G ) )
assume A1: ( 1 <= i & i <= (len s1) - 1 ) ; :: thesis: ( s1 . i is strict StableSubgroup of G & s1 . (i + 1) is strict StableSubgroup of G )
A2: 0 + i <= 1 + i by XREAL_1:8;
then A3: 1 <= i + 1 by A1, XXREAL_0:2;
A4: i + 1 <= ((len s1) - 1) + 1 by A1, XREAL_1:8;
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 A2, A4, XXREAL_0:2;
then i in Seg (len s1) by A1, FINSEQ_1:3;
then i in dom s1 by FINSEQ_1:def 3;
then A6: s1 . i is Element of the_stable_subgroups_of G by FINSEQ_2:13;
A7: s1 . (i + 1) is Element of the_stable_subgroups_of G by A5, FINSEQ_2:13;
thus s1 . i is strict StableSubgroup of G by A6, Def11; :: thesis: s1 . (i + 1) is strict StableSubgroup of G
thus s1 . (i + 1) is strict StableSubgroup of G by A7, Def11; :: thesis: verum