
theorem Th11:
  for n be non zero Nat holds Absval (Bin1 n) = 1
proof
  defpred P[Nat] means Absval (Bin1 $1) = 1;
A1: for n be non zero Nat st P[n] holds P[n+1]
  proof
    let n be non zero Nat;
    assume
A2: Absval (Bin1 n) = 1;
    thus Absval (Bin1 (n+1)) = Absval (Bin1 n ^ <*FALSE*>) by BINARI_2:7
      .= Absval Bin1 n + IFEQ(FALSE,FALSE,0,2 to_power n) by BINARITH:20
      .= Absval Bin1 n + 0 by FUNCOP_1:def 8
      .= 1 by A2;
  end;
A3: P[1] by Th10,BINARITH:16;
  thus for n being non zero Nat holds P[n] from NAT_1:sch 10(A3,A1);
end;
