reserve i,j,k,n for Nat;
reserve x,y,z for Tuple of n, BOOLEAN;

theorem Th6:
  for n,x holds x 'xor' x = ZERO(n)
proof
  let n;
  let x be Tuple of n, BOOLEAN;
A1: len (x 'xor' x) = n by CARD_1:def 7;
  then
A2: dom (x 'xor' x) = Seg n by FINSEQ_1:def 3;
A3: now
    let j be Nat;
    assume
A4: j in dom (x 'xor' x);
A5: ZERO(n).j = FALSE;
    thus (x 'xor' x).j = (x 'xor' x)/.j by A4,PARTFUN1:def 6
      .= (x/.j) 'xor' (x/.j) by A2,A4,Def2
      .= ZERO(n).j by A5,XBOOLEAN:138;
  end;
  len (ZERO(n)) = n by CARD_1:def 7;
  hence thesis by A1,A3,FINSEQ_2:9;
end;
