
theorem
  for n,k being Element of NAT st k < n for x,y being FinSequence holds
  (k+1,n)-BitSubtracterOutput(x,y) =
  BitSubtracterOutput(x .(k+1), y.(k+1), k-BitBorrowOutput(x,y))
proof
  let n,k be Element of NAT such that
A1: k < n;
  let x,y be FinSequence;
A2: k+1 >= 1 by NAT_1:11;
  k+1 <= n by A1,NAT_1:13;
  then ex i being Element of NAT st ( k+1 = i+1)&( (k+1,n)
  -BitSubtracterOutput(x,y) = BitSubtracterOutput(x .(k+1), y.( k+1), i
  -BitBorrowOutput(x,y))) by A2,Def4;
  hence thesis;
end;
