
theorem Th33:
  for L be add-associative right_zeroed well-unital
  right_complementable right-distributive non empty doubleLoopStr
  for p be sequence of L holds p*'1_.(L) = p
proof
  let L be add-associative right_zeroed well-unital right_complementable
  right-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: (p*'1_.(L)).i = Sum r and
A3: for k be Element of NAT st k in dom r holds r.k = p.(k-'1) * (1_.(
    L)).(i+1-'k) by Def9;
a1: r <> {} by A1;
    i+1 in Seg len r by A1,FINSEQ_1:4;
    then
A4: i+1 in dom r by FINSEQ_1:def 3;
    now
      let k be Element of NAT;
      assume k in dom Del(r,i+1);
      then
A5:   k in Seg len Del(r,i+1) by FINSEQ_1:def 3;
      then k in Seg i by A1,PRE_POLY:12;
      then
A6:   k <= i by FINSEQ_1:1;
      then
A7:   k < i+1 by NAT_1:13;
A8:   i+1-k <> 0 by A6,NAT_1:13;
      i+1-k >= 0 by A7,XREAL_1:48;
      then
A9:   i+1-'k <> 0 by A8,XREAL_0:def 2;
      1 <= k by A5,FINSEQ_1:1;
      then k in Seg (i+1) by A7,FINSEQ_1:1;
      then
A10:  k in dom r by A1,FINSEQ_1:def 3;
      thus Del(r,i+1).k = r.k by A7,FINSEQ_3:110
        .= p.(k-'1) * (1_.(L)).(i+1-'k) by A3,A10
        .= p.(k-'1) * 0.L by A9,Th28
        .= 0.L;
    end;
    then
A11: Sum Del(r,i+1) = 0.L by Th1;
    r = Del(r,i+1) ^ <*r.(i+1)*> by A1,a1,PRE_POLY:13
      .= Del(r,i+1) ^ <*r/.(i+1)*> by A4,PARTFUN1:def 6;
    then
A12: Sum r = Sum Del(r,i+1) + Sum <*r/.(i+1)*> by RLVECT_1:41
      .= Sum Del(r,i+1) + (r/.(i+1)) by RLVECT_1:44;
    r/.(i+1) = r.(i+1) by A4,PARTFUN1:def 6
      .= p.(i+1-'1) * (1_.(L)).(i+1-'(i+1)) by A3,A4
      .= p.i * (1_.(L)).(i+1-'(i+1)) by NAT_D:34
      .= p.i * (1_.(L)).0 by XREAL_1:232
      .= (p.i) * 1_L by Th28
      .= p.i;
    hence (p*'1_.(L)).i = p.i by A2,A12,A11,RLVECT_1:4;
  end;
  hence thesis;
end;
