reserve i,j,k,n,m for Nat,
        b,b1,b2 for bag of n;

theorem Th4:
  for b being bag of (n+1) holds
     b = (0,n)-cut b bag_extend (b.n)
proof
  let b be bag of (n+1);
  set C=(0,n)-cut b, B= C bag_extend (b.n);
A1: n-'0 = n by NAT_D:40;
  then
A2: B|n = C & B.n = b.n by HILBASIS:def 1;
A3: C=b|n by Th3,NAT_1:11;
A4:dom b = n+1 = dom B by A1,PARTFUN1:def 2;
  for x be object st x in n+1 holds B.x = b.x
  proof
    let x be object;
    assume x in n+1;
    then x in Segm (n+1);
    then x in (Segm n)\/{n} by AFINSQ_1:2;
    then x in n or x=n by ZFMISC_1:136;
    then B.x = C.x = b.x or B.x = b.x by FUNCT_1:49,A2,A3;
    hence thesis;
  end;
  hence thesis by FUNCT_1:2,A4;
end;
