
theorem Th25:
  for n be Nat holds n-BinarySequence 0 = 0*n
proof
  let n be Nat;
  0*n in BOOLEAN* by Th4;
  then
 0*n is FinSequence of BOOLEAN by FINSEQ_1:def 11;
  then reconsider F = 0*n as Tuple of n,BOOLEAN;
  now
    let i be Nat;
    assume
A1: i in Seg n;
    len (n-BinarySequence 0) = n by CARD_1:def 7;
    then i in dom (n-BinarySequence 0) by A1,FINSEQ_1:def 3;
    hence (n-BinarySequence 0).i = (n-BinarySequence 0)/.i by PARTFUN1:def 6
      .= IFEQ((0 div 2 to_power (i-'1)) mod 2,0,FALSE,TRUE) by A1,Def1
      .= IFEQ(0 mod 2,0,FALSE,TRUE)
      .= IFEQ(0,0,FALSE,TRUE) by NAT_D:26
      .= 0 by FUNCOP_1:def 8
      .= F.i by A1,FUNCOP_1:7;
  end;
  hence thesis by FINSEQ_2:119;
end;
