
theorem Th11:
  for n being Ordinal, L being right_zeroed add-associative
  right_complementable well-unital distributive non trivial
doubleLoopStr, p being Polynomial of n,L, b being bag of n st Support p = {b}
  for x being Function of n, L holds eval(p,x) = p.b * eval(b,x)
proof
  let n be Ordinal, L be right_zeroed add-associative right_complementable
well-unital distributive non trivial doubleLoopStr, p be Polynomial
  of n,L, b be bag of n;
  reconsider sp = Support p as finite Subset of Bags n;
  set sg = SgmX(BagOrder n, sp);
A1: b in Bags n by PRE_POLY:def 12;
A2: dom p = Bags n by FUNCT_2:def 1;
A3: BagOrder n linearly_orders sp by Th10;
  assume Support p = {b};
  then
A4: rng sg = {b} by A3,PRE_POLY:def 2;
  then
A5: b in rng sg by TARSKI:def 1;
  then
A6: 1 in dom sg by FINSEQ_3:31;
  then
A7: sg/.1 = sg.1 by PARTFUN1:def 6;
A8: for u being object holds u in dom sg implies u in {1}
  proof
    let u be object;
    assume
A9: u in dom sg;
    assume
A10: not u in {1};
    reconsider u as Element of NAT by A9;
    sg/.u = sg.u by A9,PARTFUN1:def 6;
    then
A11: sg/.u in rng sg by A9,FUNCT_1:def 3;
A12: u <> 1 by A10,TARSKI:def 1;
A13: 1 < u
    proof
      consider k being Nat such that
A14:  dom sg = Seg k by FINSEQ_1:def 2;
      Seg k = {l where l is Nat : 1 <= l & l <= k} by FINSEQ_1:def 1;
      then
      ex m9 being Nat st m9 = u & 1 <= m9 & m9 <= k by A9,A14;
      hence thesis by A12,XXREAL_0:1;
    end;
    sg/.1 = sg.1 by A5,A9,FINSEQ_3:31,PARTFUN1:def 6;
    then sg/.1 in rng sg by A6,FUNCT_1:def 3;
    then sg/.1 = b by A4,TARSKI:def 1
      .= sg/.u by A4,A11,TARSKI:def 1;
    hence thesis by A3,A6,A9,A13,PRE_POLY:def 2;
  end;
  for u being object holds u in {1} implies u in dom sg by A6,TARSKI:def 1;
  then
A15: dom sg = Seg 1 by A8,FINSEQ_1:2,TARSKI:2;
  then
A16: len sg = 1 by FINSEQ_1:def 3;
A17: sg.1 in rng sg by A6,FUNCT_1:def 3;
  then sg.1 = b by A4,TARSKI:def 1;
  then 1 in dom (p * sg) by A6,A1,A2,FUNCT_1:11;
  then
A18: (p * sg)/.1 = (p * sg).1 by PARTFUN1:def 6
    .= p.(sg.1) by A6,FUNCT_1:13
    .= p.b by A4,A17,TARSKI:def 1;
  1 in dom sg by A15,FINSEQ_1:2,TARSKI:def 1;
  then
A19: sg/.1 in rng sg by A7,FUNCT_1:def 3;
  let x be Function of n, L;
  consider y being FinSequence of the carrier of L such that
A20: len y = len SgmX(BagOrder n, Support p) and
A21: eval(p,x) = Sum y and
A22: for i being Element of NAT st 1 <= i & i <= len y holds y/.i = (p *
  SgmX(BagOrder n, Support p))/.i * eval(((SgmX(BagOrder n, Support p))/.i),x)
  by Def2;
  y.1 = y/.1 by A20,A16,FINSEQ_4:15
    .= (p * sg)/.1 * eval((sg/.1),x) by A20,A22,A16
    .= (p * sg)/.1 * eval(b,x) by A4,A19,TARSKI:def 1;
  then y = <* p.b * eval(b,x) *> by A20,A16,A18,FINSEQ_1:40;
  hence thesis by A21,RLVECT_1:44;
end;
