theorem Th112:
  1<=i & i<=len s1-1 & H1=s1.i & H2=s1.(i+1) implies H2 is normal
  StableSubgroup of H1
proof
  assume that
A1: 1<=i and
A2: i<=len s1-1;
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;
  then
A6: i in dom s1 by FINSEQ_1:def 3;
  assume H1=s1.i & H2=s1.(i+1);
  hence thesis by A5,A6,Def28;
end;
