reserve i,n,m,k,x for Nat,
  i1,i2 for Integer;
reserve a for Tuple of n,k-SD;
reserve aSub for Tuple of n,k-SD_Sub;

theorem Th14:
  for x be Integer, k be Nat st 2 <= k holds SDSub_Add_Carry(x,k) in k-SD_Sub_S
proof
  let x be Integer, k be Nat;
A1: SDSub_Add_Carry(x,k) <= 1 by Th12;
  assume k >= 2;
  then k > 1 by XXREAL_0:2;
  then k - 1 > 1 - 1 by XREAL_1:14;
  then
A2: k -' 1 > 0 by XREAL_0:def 2;
  then 2 to_power (k -' 1) > 1 by POWER:35;
  then
A3: -Radix(k-'1) <= -1 by XREAL_1:24;
  -1 <= SDSub_Add_Carry(x,k) by Th12;
  then
A4: SDSub_Add_Carry(x,k) >= -Radix(k-'1) by A3,XXREAL_0:2;
  Radix(k-'1) - 1 >= 1 by A2,INT_1:52,POWER:35;
  then
A5: SDSub_Add_Carry(x,k) <= Radix(k-'1) - 1 by A1,XXREAL_0:2;
  SDSub_Add_Carry(x,k) is Element of INT by INT_1:def 2;
  hence thesis by A5,A4;
end;
