
theorem
  for X being set, L being non empty ZeroStr, a,b being Element of L
  holds a |(X,L) = b |(X,L) iff a = b
proof
  let X be set, L be non empty ZeroStr, a,b be Element of L;
  set m = 0_(X,L)+*(EmptyBag X,a);
  reconsider m as Function of Bags X, the carrier of L;
  reconsider m as Function of Bags X, L;
  reconsider m as Series of X, L;
  set k = 0_(X,L)+*(EmptyBag X,b);
  reconsider k as Function of Bags X, the carrier of L;
  reconsider k as Function of Bags X, L;
  reconsider k as Series of X, L;
A1: EmptyBag X in dom(EmptyBag X .--> a) by TARSKI:def 1;
A2: EmptyBag X in dom(EmptyBag X .--> b) by TARSKI:def 1;
  dom(0_(X,L)) = dom((Bags X) --> 0.L) by POLYNOM1:def 8
    .= Bags X;
  then
A3: k.(EmptyBag X) = (0_(X,L)+*(EmptyBag X .--> b)).(EmptyBag X) by
FUNCT_7:def 3
    .= (EmptyBag X .--> b).(EmptyBag X) by A2,FUNCT_4:13
    .= b by FUNCOP_1:72;
  dom(0_(X,L)) = dom((Bags X) --> 0.L) by POLYNOM1:def 8
    .= Bags X;
  then m.(EmptyBag X) = (0_(X,L)+*(EmptyBag X .--> a)).(EmptyBag X) by
FUNCT_7:def 3
    .= (EmptyBag X .--> a).(EmptyBag X) by A1,FUNCT_4:13
    .= a by FUNCOP_1:72;
  hence thesis by A3;
end;
