reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem Th67:
  for n being Ordinal, b, b1, b2 being bag of n st b = b1+b2 ex i
  being Element of NAT st i in dom decomp b & (decomp b)/.i = <*b1, b2*>
proof
  let n be Ordinal, b, b1, b2 be bag of n;
  consider S being non empty finite Subset of Bags n such that
A1: divisors b = SgmX(BagOrder n, S) and
A2: for p being bag of n holds p in S iff p divides b by Def15;
A3: BagOrder n linearly_orders S by Lm4,ORDERS_1:38;
  assume
A4: b = b1+b2;
  then b1 divides b by Th49;
  then b1 in S by A2;
  then b1 in rng divisors b by A1,A3,Def2;
  then consider i being Element of NAT such that
A5: i in dom divisors b and
A6: (divisors b)/.i= b1 by PARTFUN2:2;
  take i;
  thus i in dom decomp b by A5,Def16;
  then (decomp b)/.i = <*b1, b-'b1*> by A6,Def16;
  hence thesis by A4,Th47;
end;
