
theorem Th18:
  for L be well-unital add-associative right_zeroed
right_complementable associative non degenerated non empty doubleLoopStr for
  x be Element of L holds eval(1_.(L),x) = 1.L
proof
  let L be well-unital add-associative right_zeroed right_complementable
  associative non degenerated non empty doubleLoopStr;
  let x be Element of L;
  consider F be FinSequence of the carrier of L such that
A1: eval(1_.(L),x) = Sum F and
A2: len F = len 1_.(L) and
A3: for n be Element of NAT st n in dom F holds F.n = (1_.(L)).(n-'1) *
  (power L).(x,n-'1) by Def2;
A4: len F = 1 by A2,Th4;
  then 1 in Seg len F by FINSEQ_1:1;
  then 1 in dom F by FINSEQ_1:def 3;
  then F.1 = (1_.(L)).(1-'1) * (power L).(x,1-'1) by A3
    .= (1_.(L)).(0) * (power L).(x,1-'1) by XREAL_1:232
    .= 1.L * (power L).(x,1-'1) by POLYNOM3:30
    .= (power L).(x,1-'1)
    .= (power L).(x,0) by XREAL_1:232
    .= 1_L by GROUP_1:def 7
    .= 1.L;
  then F = <*1.L*> by A4,FINSEQ_1:40;
  hence thesis by A1,RLVECT_1:44;
end;
