
theorem Th23:
  for R being right_zeroed unital non empty doubleLoopStr, a,b
  being Element of R, n being Nat holds ((a,b) In_Power n).1 = a|^n
proof
  reconsider m = 1 - 1 as Element of NAT by NEWTON:19;
  let R be right_zeroed unital non empty doubleLoopStr, a,b be Element of R,
  n be Nat;
  reconsider l = n - m as Nat;
  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;
  0 + 1 <= n + 1 by XREAL_1:6;
  then
A2: 1 in dom(((a,b) In_Power n)) by A1,FINSEQ_1:1;
  hence ((a,b) In_Power n).1 = ((a,b) In_Power n)/.1 by PARTFUN1:def 6
    .= (n choose 0) * a|^l * b|^m by A2,Def7
    .= 1 * a|^n * b|^0 by NEWTON:19
    .= a|^n * b|^0 by Th13
    .= a|^n * 1_R by Th8
    .= a|^n by GROUP_1:def 4;
end;
