
theorem Th6:
  for n be non zero Nat for F be Tuple of n,BOOLEAN st F = 0*n
  holds Absval F = 0
proof
  defpred P[Nat] means for F be Tuple of $1,BOOLEAN st F = 0*$1 holds Absval F
  = 0;
A1: for n be non zero Nat st P[n] holds P[n+1]
  proof
    let n be non zero Nat;
    assume
A2: for F be Tuple of n,BOOLEAN st F = 0*n holds Absval F = 0;
    let F be Tuple of n+1,BOOLEAN;
    0*n in BOOLEAN* by Th4;
    then
 0*n is FinSequence of BOOLEAN by FINSEQ_1:def 11;
    then reconsider F1 = 0*n as Tuple of n,BOOLEAN;
    assume F = 0*(n+1);
    hence Absval F = Absval(F1 ^ <*FALSE*>) by FINSEQ_2:60
      .= Absval(F1) + IFEQ(FALSE,FALSE,0,2 to_power n) by BINARITH:20
      .= 0 + IFEQ(FALSE,FALSE,0,2 to_power n) by A2
      .= 0 by FUNCOP_1:def 8;
  end;
A3: P[1] by BINARITH:15,FINSEQ_2:59;
  thus for n being non zero Nat holds P[n] from NAT_1:sch 10(A3,A1);
end;
