
theorem Th12:
  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(0_(n,L),x) = 0.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 0p = 0_(n,L);
  consider y being FinSequence of the carrier of L such that
A1: len y = len SgmX(BagOrder n, Support 0p) and
A2: Sum y = eval(0p,x) and
  for i being Element of NAT st 1 <= i & i <= len y holds y/.i = (0p *
SgmX(BagOrder n, Support 0p))/.i * eval(((SgmX(BagOrder n, Support 0p))/.i),x)
  by Def2;
  Support 0p = {}
  proof
    set u = the Element of Support 0p;
    assume Support 0p <> {};
    then
A3: u in Support 0p;
    then
A4: u is Element of Bags n;
    0p.u <> 0.L by A3,POLYNOM1:def 4;
    hence thesis by A4,POLYNOM1:22;
  end;
  then SgmX(BagOrder n, Support 0p) = {} by Th10,PRE_POLY:76;
  then y = <*>the carrier of L by A1;
  hence thesis by A2,RLVECT_1:43;
end;
