
theorem Th12:
  for X being set, L being unital non trivial
doubleLoopStr, x be Element of X holds 1_1(x,L).UnitBag x = 1_L & for b being
  bag of X st b <> UnitBag x holds 1_1(x,L).b = 0.L
proof
  let X be set, L be unital non trivial doubleLoopStr, x be
  Element of X;
  dom (0_(X,L)) = dom ((Bags X) --> 0.L) by POLYNOM1:def 8
    .= Bags X;
  hence 1_1(x,L).UnitBag x = 1_L by FUNCT_7:31;
  let b be bag of X;
  assume b <> UnitBag x;
  hence 1_1(x,L).b = (0_(X,L)).b by FUNCT_7:32
    .= 0.L by POLYNOM1:22;
end;
