
theorem Th18:
  for X being set, L being non empty multLoopStr_0, a being
  Element of L holds (a |(X,L)).EmptyBag X = a & for b being bag of X st b <>
  EmptyBag X holds (a |(X,L)).b = 0.L
proof
  let n be set, L be non empty multLoopStr_0, a be Element of L;
  set Z = 0_(n,L);
A1: Z = (Bags n --> 0.L) by POLYNOM1:def 8;
  then dom Z = Bags n;
  hence (a |(n,L)).EmptyBag n = a by FUNCT_7:31;
  let b be bag of n;
A2: b in Bags n by PRE_POLY:def 12;
  assume b <> EmptyBag n;
  hence (a |(n,L)).b = Z.b by FUNCT_7:32
    .= 0.L by A1,A2,FUNCOP_1:7;
end;
