
theorem Th24:
  for n be non zero Nat for z be Tuple of n,BOOLEAN st z = 0*n
  holds 'not' z + Bin1 n = z
proof
  let n be non zero Nat;
  let z be Tuple of n,BOOLEAN;
  assume
A1: z = 0*n;
  then
A2: add_ovfl('not' z,Bin1 n) = TRUE by Th23;
  Absval ('not' z + Bin1 n) = Absval ('not' z + Bin1 n) + 2 to_power n - 2
  to_power n
    .= Absval ('not' z + Bin1 n) + IFEQ(add_ovfl('not' z,Bin1 n),FALSE, 0,2
  to_power (n)) - 2 to_power n by A2,FUNCOP_1:def 8
    .= Absval 'not' z + Absval Bin1 n - 2 to_power n by BINARITH:21
    .= - Absval(z) + 2 to_power n - 1 + Absval Bin1 n - 2 to_power n by
BINARI_2:13
    .= - Absval(z) + 2 to_power n - 1 + 1 - 2 to_power n by Th11
    .= -0 by A1,Th6
    .= Absval z by A1,Th6;
  hence thesis by Th2;
end;
