
theorem
  for n be Nat for y be Tuple of n,BOOLEAN st y = 0*n holds Rev 'not' y
  = 'not' y
proof
  let n be Nat;
  let y be Tuple of n,BOOLEAN;
  assume
A1: y = 0*n;
A2: now
    let k be Nat;
    assume
A3: k in dom 'not' y;
    then k in Seg len 'not' y by FINSEQ_1:def 3;
    then
A4: k in Seg n by CARD_1:def 7;
    then n - k + 1 in Seg n by FINSEQ_5:2;
    then
A5: len 'not' y - k + 1 in Seg n by CARD_1:def 7;
    thus (Rev 'not' y).k = ('not' y).(len 'not' y - k + 1) by A3,FINSEQ_5:58
      .= (n |-> 1).(len 'not' y - k + 1) by A1,Th5
      .= 1 by A5,FUNCOP_1:7
      .= (n |-> 1).k by A4,FUNCOP_1:7
      .= ('not' y).k by A1,Th5;
  end;
  dom Rev 'not' y = Seg len Rev 'not' y by FINSEQ_1:def 3
    .= Seg len 'not' y by FINSEQ_5:def 3
    .= dom 'not' y by FINSEQ_1:def 3;
  hence thesis by A2,FINSEQ_1:13;
end;
