
theorem Th32:
  for n be non zero Nat for k be Nat st k + 1 < 2 to_power n
  holds add_ovfl(n-BinarySequence k,Bin1 n) = FALSE
proof
  let n be non zero Nat;
  let k be Nat;
  assume
A1: k + 1 < 2 to_power n;
  then
A2: k < 2 to_power n by NAT_1:13;
  0*n in BOOLEAN* by Th4;
  then
 0*n is FinSequence of BOOLEAN by FINSEQ_1:def 11;
  then reconsider y = 0*n as Tuple of n,BOOLEAN;
  k < 2 to_power n - 1 by A1,XREAL_1:20;
  then n-BinarySequence k <> 'not' y by A2,Lm4;
  then add_ovfl(n-BinarySequence k,Bin1 n) <> TRUE by Th23;
  hence thesis by XBOOLEAN:def 3;
end;
