
theorem Th13:
  for n being Ordinal, L being right_zeroed add-associative
  right_complementable well-unital distributive non trivial
  doubleLoopStr, x being Function of n, L holds eval(1_(n,L),x) = 1.L
proof
  let n be Ordinal, L be right_zeroed add-associative right_complementable
  well-unital distributive non trivial doubleLoopStr, x be Function
  of n, L;
  set 1p = 1_(n,L);
A1: for u being object holds u in {EmptyBag n} implies u in Support 1p
  proof
    let u be object;
    assume
A2: u in {EmptyBag n};
    then
A3: u = EmptyBag n by TARSKI:def 1;
    reconsider u as Element of Bags n by A2,TARSKI:def 1;
    1p.u = 1.L by A3,POLYNOM1:25;
    then 1p.u <> 0.L;
    hence thesis by POLYNOM1:def 4;
  end;
  reconsider s1p = Support 1p as finite Subset of Bags n;
  set sg1p = SgmX(BagOrder n, s1p);
A4: BagOrder n linearly_orders Support 1p by Th10;
  for u being object holds u in Support 1p implies u in {EmptyBag n}
  proof
    let u be object;
    assume
A5: u in Support 1p;
    then reconsider u as Element of Bags n;
    1p.u <> 0.L by A5,POLYNOM1:def 4;
    then u = EmptyBag n by POLYNOM1:25;
    hence thesis by TARSKI:def 1;
  end;
  then
A6: Support 1p = {EmptyBag n} by A1,TARSKI:2;
  then
A7: rng sg1p = {EmptyBag n} by A4,PRE_POLY:def 2;
  then
A8: EmptyBag n in rng sg1p by TARSKI:def 1;
  then
A9: 1 in dom sg1p by FINSEQ_3:31;
  then sg1p/.1 = sg1p.1 by PARTFUN1:def 6;
  then sg1p/.1 in rng sg1p by A9,FUNCT_1:def 3;
  then
A10: sg1p/.1 = EmptyBag n by A7,TARSKI:def 1;
A11: for u being object holds u in dom sg1p implies u in {1}
  proof
    let u be object;
    assume
A12: u in dom sg1p;
    assume
A13: not u in {1};
    reconsider u as Element of NAT by A12;
    sg1p/.u = sg1p.u by A12,PARTFUN1:def 6;
    then
A14: sg1p/.u in rng sg1p by A12,FUNCT_1:def 3;
A15: u <> 1 by A13,TARSKI:def 1;
A16: 1 < u
    proof
      consider k being Nat such that
A17:  dom sg1p = Seg k by FINSEQ_1:def 2;
      Seg k = {m where m is Nat : 1 <= m & m <= k} by FINSEQ_1:def 1;
      then
      ex m9 being Nat st m9 = u & 1 <= m9 & m9 <= k by A12,A17;
      hence thesis by A15,XXREAL_0:1;
    end;
    sg1p/.1 = sg1p.1 by A8,A12,FINSEQ_3:31,PARTFUN1:def 6;
    then sg1p/.1 in rng sg1p by A9,FUNCT_1:def 3;
    then sg1p/.1 = EmptyBag n by A7,TARSKI:def 1
      .= sg1p/.u by A7,A14,TARSKI:def 1;
    hence thesis by A4,A9,A12,A16,PRE_POLY:def 2;
  end;
A18: dom 1p = Bags n by FUNCT_2:def 1;
A19: 1 in dom sg1p by A8,FINSEQ_3:31;
A20: sg1p.1 in rng sg1p by A9,FUNCT_1:def 3;
  then sg1p.1 in {EmptyBag n} by A6,A4,PRE_POLY:def 2;
  then sg1p.1 = EmptyBag n by TARSKI:def 1;
  then 1 in dom (1p * sg1p) by A19,A18,FUNCT_1:11;
  then
A21: (1p * sg1p)/.1 = (1p * sg1p).1 by PARTFUN1:def 6
    .= 1p.(sg1p.1) by A9,FUNCT_1:13
    .= 1p.(EmptyBag n) by A7,A20,TARSKI:def 1
    .= 1.L by POLYNOM1:25;
  consider y being FinSequence of the carrier of L such that
A22: len y = len sg1p and
A23: Sum y = eval(1p,x) and
A24: for i being Element of NAT st 1 <= i & i <= len y holds y/.i = (1p
  * sg1p)/.i * eval(((sg1p)/.i),x) by Def2;
  for u being object holds u in {1} implies u in dom sg1p by A9,TARSKI:def 1;
  then dom sg1p = Seg 1 by A11,FINSEQ_1:2,TARSKI:2;
  then
A25: len sg1p = 1 by FINSEQ_1:def 3;
  then y.1 = y/.1 by A22,FINSEQ_4:15
    .= (1p * sg1p)/.1 * eval(((sg1p)/.1),x) by A25,A22,A24
    .= (1p * sg1p)/.1 * 1.L by A10,Th6
    .= (1p * sg1p)/.1;
  then y = <* 1.L *> by A25,A22,A21,FINSEQ_1:40;
  hence thesis by A23,RLVECT_1:44;
end;
