
theorem Th5:
  for n being Element of NAT, p being Element of n-tuples_on BOOLEAN holds (n
  -BinarySequence 0) '&' p = n-BinarySequence 0
proof
  let n be Element of NAT, p be Element of n-tuples_on BOOLEAN;
  set B = n-BinarySequence 0;
  now
    let x be object;
A1: dom B = Seg n by Lm1;
A2: dom (B '&' p) = Seg n by Lm1;
    hence dom (B '&' p) = dom B by Lm1;
    let x be object;
    assume
A3: x in dom (B '&' p);
A4: B = 0*n by BINARI_3:25
      .= n |-> 0 by EUCLID:def 4;
    then B.x = 0;
    then B/.x = FALSE by A2,A3,A1,PARTFUN1:def 6;
    hence (B '&' p).x = FALSE '&' (p/.x) by A2,A3,Def5
      .= B.x by A4;
  end;
  hence thesis by FUNCT_1:2;
end;
