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

theorem Th7:
  for n,x,y holds x 'xor' y = y 'xor' x
proof
  let n;
  let x,y be Tuple of n, BOOLEAN;
A1: len (x 'xor' y) = n by CARD_1:def 7;
  then
A2: dom (x 'xor' y) = Seg n by FINSEQ_1:def 3;
A3: len (y 'xor' x) = n by CARD_1:def 7;
  now
    let j be Nat;
    assume
A4: j in dom (x 'xor' y);
    then
A5: j in dom (y 'xor' x) by A3,A2,FINSEQ_1:def 3;
    thus (x 'xor' y).j = (x 'xor' y)/.j by A4,PARTFUN1:def 6
      .= (y/.j) 'xor' (x/.j) by A2,A4,Def2
      .= (y 'xor' x)/.j by A2,A4,Def2
      .= (y 'xor' x).j by A5,PARTFUN1:def 6;
  end;
  hence thesis by A1,A3,FINSEQ_2:9;
end;
