
theorem Th25:
  for n being set, L being right_unital non empty multLoopStr_0
  holds (1_(n,L)).EmptyBag n = 1.L & for b being bag of n st b <> EmptyBag n
  holds (1_(n,L)).b = 0.L
proof
  let n be set, L be right_unital non empty multLoopStr_0;
  set Z = 0_(n,L);
  dom Z = Bags n;
  hence (1_(n,L)).EmptyBag n = 1.L by FUNCT_7:31;
  let b be bag of n;
A1: b in Bags n by PRE_POLY:def 12;
  assume b <> EmptyBag n;
  hence (1_(n,L)).b = Z.b by FUNCT_7:32
    .= 0.L by A1,FUNCOP_1:7;
end;
