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

theorem Th3:
  k-SD_Sub_S c= (k+1)-SD_Sub_S
proof
  let e be object;
  assume e in k-SD_Sub_S;
  then consider g being Element of INT such that
A1: e = g and
A2: g >= -Radix(k-'1) and
A3: g <= Radix(k-'1) - 1;
  k-'1 <= k by NAT_D:44;
  then
A4: 2 to_power (k-'1) <= 2 to_power k by PRE_FF:8;
  then Radix(k-'1) - 1 <= Radix(k) - 1 by XREAL_1:9;
  then Radix(k-'1) - 1 <= Radix(k + 1 -' 1) - 1 by NAT_D:34;
  then
A5: g <= Radix(k + 1 -' 1) - 1 by A3,XXREAL_0:2;
  -Radix(k-'1) >= -Radix(k) by A4,XREAL_1:24;
  then -Radix(k-'1) >= -Radix(k + 1 -' 1) by NAT_D:34;
  then g >= -Radix(k + 1 -' 1) by A2,XXREAL_0:2;
  hence thesis by A1,A5;
end;
