
theorem Th5:
  for n be Nat for y be Tuple of n,BOOLEAN st y = 0*n holds 'not' y = n |-> 1
proof
  let n be Nat;
  let y be Tuple of n,BOOLEAN;
  assume
A1: y = 0*n;
A2: now
A3: len y = n by CARD_1:def 7;
    let i be Nat;
    assume that
A4: 1 <= i and
A5: i <= len 'not' y;
A6: len 'not' y = n by CARD_1:def 7;
    then
A7: i in Seg n by A4,A5,FINSEQ_1:1;
    then
A8: y.i = FALSE by A1,FUNCOP_1:7;
    thus ('not' y).i = ('not' y)/.i by A4,A5,FINSEQ_4:15
      .= 'not' (y/.i) by A7,BINARITH:def 1
      .= 'not' FALSE by A4,A5,A6,A3,A8,FINSEQ_4:15
      .= (n |-> 1).i by A7,FUNCOP_1:7;
  end;
  len 'not' y = n by CARD_1:def 7
    .= len (n |-> 1) by CARD_1:def 7;
  hence thesis by A2,FINSEQ_1:14;
end;
