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 Th15:
  2 <= k & i1 in k-SD implies SDSub_Add_Data(i1,k) +
  SDSub_Add_Carry(i2,k) in k-SD_Sub
proof
A1: SDSub_Add_Carry(i2,k) >= -1 by Th12;
  assume
A2: 2 <= k & i1 in k-SD;
  then SDSub_Add_Data(i1,k) >= -Radix(k-'1) by Th13;
  then
A3: SDSub_Add_Data(i1,k) + SDSub_Add_Carry(i2,k) >= -Radix(k-'1) + - 1 by A1,
XREAL_1:7;
A4: SDSub_Add_Carry(i2,k) <= 1 by Th12;
  SDSub_Add_Data(i1,k) <= Radix(k-'1) - 1 by A2,Th13;
  then
A5: SDSub_Add_Data(i1,k) + SDSub_Add_Carry(i2,k) <= Radix(k-'1) - 1 + 1 by A4,
XREAL_1:7;
  SDSub_Add_Data(i1,k) + SDSub_Add_Carry(i2,k) is Element of INT by INT_1:def 2
;
  hence thesis by A3,A5;
end;
