
theorem Th22:
  for R being right_zeroed unital non empty doubleLoopStr, a,b
  being Element of R holds (a,b) In_Power 0 = <*1_R*>
proof
  let R be right_zeroed unital non empty doubleLoopStr, a,b being Element of
  R;
  set p = (a,b) In_Power 0;
A1: len p = 0 + 1 by Def7
    .= 1;
  then
A2: dom p = {1} by FINSEQ_1:2,def 3;
  then
A3: 1 in dom p by TARSKI:def 1;
  then consider i being Element of NAT such that
A4: i in dom p;
A5: i = 1 by A2,A4,TARSKI:def 1;
  then reconsider m = i - 1 as Element of NAT by INT_1:5;
  reconsider l = 0 - m as Element of NAT by A5;
  p.1 = p/.1 by A3,PARTFUN1:def 6
    .= (0 choose m) * a|^l * b|^m by A3,A5,Def7
    .= 1 * a|^l * b|^m by A5,NEWTON:19
    .= 1 * a|^0 * 1_R by A5,Th8
    .= 1 * 1_R * 1_R by Th8
    .= 1_R * 1_R by Th13
    .= 1_R by GROUP_1:def 4;
  hence thesis by A1,FINSEQ_1:40;
end;
