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
  for n being Ordinal, i being Nat, b, b1, b2 being bag of n st i
> 1 & i < len decomp b & (decomp b)/.i = <*b1, b2*> holds b1 <> EmptyBag n & b2
  <> EmptyBag n
proof
  let n be Ordinal, i be Nat, b, b1, b2 be bag of n such that
A1: i > 1 & i < len decomp b and
A2: (decomp b)/.i = <*b1, b2*>;
  reconsider p = (divisors b)/.i as bag of n;
A3: i in dom decomp b by A1,FINSEQ_3:25;
  then
A4: (decomp b)/.i = <*p,b-'p*> by Def16;
  then
A5: b2 = b-'p by A2,FINSEQ_1:77;
A6: dom decomp b = dom divisors b by Def16;
  then
A7: len decomp b = len divisors b by FINSEQ_3:29;
  b1 = p by A2,A4,FINSEQ_1:77;
  hence b1 <> EmptyBag n by A1,A7,Th64;
  assume b2 = EmptyBag n;
  then p = b by A6,A3,A5,Th55,Th62;
  hence contradiction by A1,A7,Th64;
end;
