theorem Th111:
  1<=i & i<=len s1-1 implies s1.i is strict StableSubgroup of G &
  s1.(i+1) is strict StableSubgroup of G
proof
  assume that
A1: 1<=i and
A2: i<=len s1-1;
A3: 0+i<=1+i by XREAL_1:6;
A4: i+1<=len s1-1+1 by A2,XREAL_1:6;
  then i<=len s1 by A3,XXREAL_0:2;
  then i in Seg len s1 by A1;
  then i in dom s1 by FINSEQ_1:def 3;
  then s1.i is Element of the_stable_subgroups_of G by FINSEQ_2:11;
  hence s1.i is strict StableSubgroup of G by Def11;
  1<=i+1 by A1,A3,XXREAL_0:2;
  then i+1 in Seg len s1 by A4;
  then i+1 in dom s1 by FINSEQ_1:def 3;
  then s1.(i+1) is Element of the_stable_subgroups_of G by FINSEQ_2:11;
  hence thesis by Def11;
end;
