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

theorem
  for n holds n-BinarySequence 0 = ZERO(n)
proof
  let n;
A1: len (n-BinarySequence 0) = n by CARD_1:def 7;
  then
A2: dom (n-BinarySequence 0) = Seg n by FINSEQ_1:def 3;
A3: len (ZERO(n)) = n by CARD_1:def 7;
  then
A4: dom ZERO(n) = Seg n by FINSEQ_1:def 3;
A5: dom (n-BinarySequence 0) = Seg n by A1,FINSEQ_1:def 3;
  now
    let j be Nat;
A6: (0 div 2 to_power(j-'1)) mod 2 = 0 mod 2
      .= 0 by NAT_D:26;
    assume
A7: j in dom (n-BinarySequence 0);
    then j in dom (ZERO(n)) by A2,FUNCOP_1:13;
    then
A8: (ZERO(n))/.j = (n |-> FALSE).j by PARTFUN1:def 6
      .= FALSE;
    thus (n-BinarySequence 0).j = (n-BinarySequence 0)/.j by A7,PARTFUN1:def 6
      .= IFEQ((0 div 2 to_power(j-'1)) mod 2,0,FALSE,TRUE) by A2,A7,
BINARI_3:def 1
      .= (ZERO(n))/.j by A6,A8,FUNCOP_1:def 8
      .= ZERO(n).j by A5,A4,A7,PARTFUN1:def 6;
  end;
  hence thesis by A1,A3,FINSEQ_2:9;
end;
