reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;

theorem Th4:
  for m be positive Nat holds card bool (Seg m\{1}) = 2|^(m-'1)
proof
  let m be positive Nat;
  defpred P[Nat] means card bool (Seg (1+$1)\{1}) = 2|^$1;
  Seg (1+0)\{1} = {} by FINSEQ_1:2;
  then card bool (Seg (1+0)\{1}) = 1 by ZFMISC_1:1,CARD_1:30;
  then A1: P[0] by NEWTON:4;
  A2:P[n] implies P[n+1]
  proof
    assume P[n];
    then card bool (Seg (1+(n+1))\{1}) = 2 * (2|^n) by HILB10_7:9
    .= 2|^(n+1) by NEWTON:6;
    hence thesis;
  end;
  A3:P[n] from NAT_1:sch 2(A1,A2);
  reconsider m1 = m-1 as Nat;
  card bool (Seg (1+m1)\{1}) = 2|^(m1) by A3;
  hence thesis;
end;
