
theorem Th24:
  for R being right_zeroed unital non empty doubleLoopStr, a,b
being Element of R, n being Nat holds ((a,b) In_Power n).(n+1) = b|^ n
proof
  let R be right_zeroed unital non empty doubleLoopStr, a,b be Element of R,
  n be Nat;
  reconsider m = n + 1 - 1 as Nat;
  reconsider l = n - m as Element of NAT by INT_1:5;
  len((a,b) In_Power n) = n + 1 by Def7;
  then
A1: dom((a,b) In_Power n) = Seg(n + 1) by FINSEQ_1:def 3;
  then
A2: l = 0 & n + 1 in dom((a,b) In_Power n) by FINSEQ_1:4;
  thus ((a,b) In_Power n).(n+1) = ((a,b) In_Power n)/.(n+1) by A1,FINSEQ_1:4
,PARTFUN1:def 6
    .= (n choose n) * a|^0 * b|^n by A2,Def7
    .= 1 * a|^0 * b|^n by NEWTON:21
    .= 1 * 1_R * b|^n by Th8
    .= 1_R * b|^n by Th13
    .= b|^n by GROUP_1:def 4;
end;
