reserve k,n,m,l,p for Nat;
reserve n0,m0 for non zero Nat;

theorem Th1:
  2|^(n+1) < 2|^(n+2) - 1
proof
  defpred P[Nat] means 2|^($1+1) < 2|^($1+2) - 1;
A1: for k be Nat st P[k] holds P[k + 1]
  proof
    let k be Nat;
    assume P[k];
    then 2|^(k+1)*2 < (2|^(k+2) - 1)*2 by XREAL_1:68;
    then 2|^((k+1)+1) < 2|^(k+2)*2 - 1*2 by NEWTON:6;
    then
A2: 2|^((k+1)+1) < 2|^((k+2)+1) - 2 by NEWTON:6;
    -2 + 2|^((k+1)+2) < -1 + 2|^((k+1)+2) by XREAL_1:8;
    hence P[k + 1] by A2,XXREAL_0:2;
  end;
  2|^1 < 2|^1*2 - 1;
  then 2|^1 < 2|^(1+1) - 1 by NEWTON:6;
  then
A3: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
