reserve i,n,m,k,x for Nat,
  i1,i2 for Integer;

theorem
  2 <= k implies k-SD_Sub c= k-SD
proof
  assume
A1: 2 <= k;
  then 1 + 1 <= k;
  then 1 <= k -' 1 by NAT_D:55;
  then
A2: Radix(k -' 1) >= 2 by Lm4;
  then Radix(k-'1) >= 1 by XXREAL_0:2;
  then -Radix(k-'1) <= -1 by XREAL_1:24;
  then
A3: Radix(k) + -Radix(k-'1) <= Radix(k) + -1 by XREAL_1:7;
  let e be object;
  assume e in k-SD_Sub;
  then consider g being Element of INT such that
A4: e = g and
A5: g >= -Radix(k-'1) - 1 and
A6: g <= Radix(k-'1);
  Radix(k -' 1) + Radix(k -' 1) >= Radix(k -' 1) + 2 by A2,XREAL_1:6;
  then Radix(k) + 0 >= (Radix(k -' 1) + 1) + 1 by A1,Lm1,XXREAL_0:2;
  then Radix(k) - 1 >= (Radix(k -' 1) + 1) - 0 by XREAL_1:21;
  then -(Radix(k -' 1) + 1) >= -(Radix(k) - 1) by XREAL_1:24;
  then
A7: g >= -Radix(k) + 1 by A5,XXREAL_0:2;
  Radix(k) + 0 = Radix(k-'1) + Radix(k-'1) by A1,Lm1,XXREAL_0:2;
  then g <= Radix(k) - 1 by A6,A3,XXREAL_0:2;
  hence thesis by A4,A7;
end;
