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 Th12:
  for x be Integer, k be Nat holds -1 <= SDSub_Add_Carry(x,k) &
  SDSub_Add_Carry(x,k) <= 1
proof
  let x be Integer, k be Nat;
  per cases;
  suppose
    x < -Radix(k-'1);
    hence thesis by Def3;
  end;
  suppose
    -Radix(k-'1) <= x & x < Radix(k-'1);
    hence thesis by Def3;
  end;
  suppose
    x >= Radix(k-'1);
    hence thesis by Def3;
  end;
end;
