
theorem Th33a:
  for L be add-associative right_zeroed well-unital
  right_complementable left-distributive non empty doubleLoopStr
  for p be sequence of L holds (1_.(L))*'p = p
proof
  let L be add-associative right_zeroed well-unital right_complementable
  left-distributive non empty doubleLoopStr;
  let p be sequence of L;
  now
    let i be Element of NAT;
    consider r be FinSequence of the carrier of L such that
A1: len r = i+1 and
A2: ((1_.(L))*'p).i = Sum r and
A3: for k be Element of NAT st k in dom r holds r.k = (1_.(L)).(k-'1)*
    p.(i+1-'k) by Def9;
A4: 1 in dom r by A1,CARD_1:27,FINSEQ_5:6;
    now
      let k be Element of NAT;
A5:   k+1 >= 1 by NAT_1:11;
      assume
A6:   k in dom Del(r,1);
      then
A7:   k<>0 by FINSEQ_3:25;
      len Del(r,1) = i by A1,A4,FINSEQ_3:109;
      then
A8:   k <= i by A6,FINSEQ_3:25;
      then k+1 <= i+ 1 by XREAL_1:6;
      then
A9:   k+1 in dom r by A1,A5,FINSEQ_3:25;
      0+1 <= k by A6,FINSEQ_3:25;
      hence Del(r,1).k = r.(k+1) by A1,A4,A8,FINSEQ_3:111
        .= (1_.(L)).(k+1-'1)*p.(i+1-'(k+1)) by A3,A9
        .= (1_.(L)).(k)*p.(i+1-'(k+1)) by NAT_D:34
        .= 0.(L)*p.(i+1-'(k+1)) by A7,Th28
        .= 0.(L);
    end;
    then
A10: Sum Del(r,1) = 0.(L) by Th1;
    r = <*r.1*>^Del(r,1) by A1,FINSEQ_5:86,CARD_1:27
      .= <*r/.1*>^Del(r,1) by A4,PARTFUN1:def 6;
    then
A11: Sum r = Sum <*r/.1*> + Sum Del(r,1) by RLVECT_1:41
      .= r/.1 + Sum Del(r,1)by RLVECT_1:44;
    r/.1 = r.1 by A4,PARTFUN1:def 6
      .= (1_.(L)).(1-'1)*p.(i+1-'1) by A3,A4
      .= (1_.(L)).(1-'1)*p.i by NAT_D:34
      .= (1_.(L)).(0)*p.i by XREAL_1:232
      .= 1_(L)*p.i by Th28
      .= p.i;
    hence ((1_.(L))*'p).i = p.i by A2,A11,A10,RLVECT_1:4;
  end;
  hence thesis;
end;
