
theorem
  for n being Element of NAT, p being Tuple of n, BOOLEAN holds 'not' (n
  -BinarySequence 0) '&' p = p
proof
  let n be Element of NAT, p be Tuple of n, BOOLEAN;
  set B = n-BinarySequence 0;
  set nB = 'not' B;
  now
    let x be set;
A1: dom B = Seg n by Lm1;
A2: dom (nB '&' p) = Seg n by Lm1;
    hence
A3: dom (nB '&' p) = dom p by Lm1;
    let x be object;
    assume
A4: x in dom (nB '&' p);
    then reconsider k=x as Element of NAT;
    B = 0*n by BINARI_3:25
      .= n |-> 0 by EUCLID:def 4;
    then B.x = 0;
    then
A5: B/.x = FALSE by A2,A4,A1,PARTFUN1:def 6;
    nB/.x = 'not' (B/.k) by A2,A4,BINARITH:def 1
      .= TRUE by A5;
    hence (nB '&' p).x = TRUE '&' (p/.x) by A2,A4,Def5
      .= p.x by A3,A4,PARTFUN1:def 6;
  end;
  hence thesis by FUNCT_1:2;
end;
