
theorem Th9:
  for L being right_zeroed add-associative right_complementable
well-unital distributive non trivial doubleLoopStr, n being Ordinal
  , p being Polynomial of n,L st Support p = {} holds p = 0_(n,L)
proof
  let L be right_zeroed add-associative right_complementable well-unital
  distributive non trivial doubleLoopStr, n be Ordinal, p be
  Polynomial of n,L such that
A1: Support p = {};
A2: for u being object st u in Bags n holds p.u = 0_(n,L).u
  proof
    let u be object;
    assume
A3: u in Bags n;
    then reconsider b = u as bag of n;
    p.b = 0.L by A1,A3,POLYNOM1:def 4;
    hence thesis by POLYNOM1:22;
  end;
A4: Bags n = dom 0_(n,L) by FUNCT_2:def 1;
  Bags n = dom p by FUNCT_2:def 1;
  hence thesis by A4,A2,FUNCT_1:2;
end;
