
theorem
  for n,k being Nat st k < n for x,y being FinSequence holds
  (k+1,n)-BitGFA1AdderOutput(x,y) =
  GFA1AdderOutput(x .(k+1), y.(k+1), k-BitGFA1CarryOutput(x,y))
proof
  let n,k be 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 Nat st ( k+1 = i+1)&( (k+1,n)-BitGFA1AdderOutput
  (x,y) = GFA1AdderOutput(x .(k+1), y.(k+1), i-BitGFA1CarryOutput(x,y))) by A2
,Def8;
  hence thesis;
end;
