
theorem Th7:
  for n being Ordinal, L being well-unital non trivial
  doubleLoopStr, u being set, b being bag of n st support b = {u} for x being
  Function of n, L holds eval(b,x) = power(L).(x.u,b.u)
proof
  let n be Ordinal, L be well-unital non trivial doubleLoopStr, u
  be set, b be bag of n;
  reconsider sb = support b as finite Subset of n;
  set sg = SgmX(RelIncl n, sb);
  assume
A1: support b = {u};
  then
A2: u in support b by TARSKI:def 1;
  let x be Function of n, L;
A3: rng x c= the carrier of L by RELAT_1:def 19;
A4: n = dom x by FUNCT_2:def 1;
  then x.u in rng x by A2,FUNCT_1:def 3;
  then reconsider xu = x.u as Element of L by A3;
A5: RelIncl n linearly_orders sb by PRE_POLY:82;
  then
A6: rng sg = {u} by A1,PRE_POLY:def 2;
  then
A7: u in rng sg by TARSKI:def 1;
  then
A8: 1 in dom sg by FINSEQ_3:31;
  then
A9: sg.1 in rng sg by FUNCT_1:def 3;
  then
A10: sg.1 = u by A6,TARSKI:def 1;
  then 1 in dom (x * sg) by A4,A8,A2,FUNCT_1:11;
  then
A11: (x * sg)/.1 = (x * sg).1 by PARTFUN1:def 6
    .= x.(sg.1) by A8,FUNCT_1:13
    .= x.u by A6,A9,TARSKI:def 1;
  dom b = n by PARTFUN1:def 2;
  then 1 in dom (b * sg) by A8,A10,A2,FUNCT_1:11;
  then
A12: (b * sg)/.1 = (b * sg).1 by PARTFUN1:def 6
    .= b.(sg.1) by A8,FUNCT_1:13
    .= b.u by A6,A9,TARSKI:def 1;
A13: power(L).(xu,b.u) = power(L).[xu,b.u];
A14: for v being object holds v in dom sg implies v in {1}
  proof
    let v be object;
    assume
A15: v in dom sg;
    assume
A16: not v in {1};
    reconsider v as Element of NAT by A15;
    sg/.v = sg.v by A15,PARTFUN1:def 6;
    then
A17: sg/.v in rng sg by A15,FUNCT_1:def 3;
A18: v <> 1 by A16,TARSKI:def 1;
A19: 1 < v
    proof
      consider k being Nat such that
A20:  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 = v & 1 <= m9 & m9 <= k by A15,A20;
      hence thesis by A18,XXREAL_0:1;
    end;
    sg/.1 = sg.1 by A7,A15,FINSEQ_3:31,PARTFUN1:def 6;
    then sg/.1 in rng sg by A8,FUNCT_1:def 3;
    then sg/.1 = u by A6,TARSKI:def 1
      .= sg/.v by A6,A17,TARSKI:def 1;
    hence thesis by A5,A8,A15,A19,PRE_POLY:def 2;
  end;
  consider y being FinSequence of the carrier of L such that
A21: len y = (len SgmX(RelIncl n, support b)) and
A22: eval(b,x) = Product y and
A23: for i being Element of NAT st 1 <= i & i <= len y holds y/.i = power
(L).((x * SgmX(RelIncl n, support b))/.i, (b * SgmX(RelIncl n, support b))/.i)
  by Def1;
  for v being object holds v in {1} implies v in dom sg by A8,TARSKI:def 1;
  then dom sg = Seg 1 by A14,FINSEQ_1:2,TARSKI:2;
  then
A24: len sg = 1 by FINSEQ_1:def 3;
  then y.1 = y/.1 by A21,FINSEQ_4:15
    .= power(L).((x * sg)/.1,(b * sg)/.1) by A21,A23,A24;
  then y = <* power(L).(x.u,b.u) *> by A21,A24,A12,A11,FINSEQ_1:40;
  hence thesis by A22,A13,GROUP_4:9;
end;
