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 Th13:
  2 <= k & i1 in k-SD implies SDSub_Add_Data(i1,k) >= -Radix(k-'1)
  & SDSub_Add_Data(i1,k) <= Radix(k-'1) - 1
proof
  assume that
A1: 2 <= k and
A2: i1 in k-SD;
A3: -Radix(k) + 1 <= i1 & 1 <= k by A1,A2,RADIX_1:13,XXREAL_0:2;
  now
    per cases;
    case
A4:   i1 < -Radix(k -' 1);
      then i1 + 1 <= -Radix(k-'1) by INT_1:7;
      then i1 <= -Radix(k-'1) - 1 by XREAL_1:19;
      then i1 + Radix(k) <= Radix(k) + ( -Radix(k-'1) - 1 ) by XREAL_1:7;
      then
A5:   i1 + Radix(k) <= Radix(k) - Radix(k-'1) - 1;
      SDSub_Add_Carry(i1,k) = -1 by A4,Def3;
      hence thesis by A3,A5,Lm2,XREAL_1:19;
    end;
    case
A6:   -Radix(k -' 1) <= i1 & i1 < Radix(k -' 1);
      then SDSub_Add_Carry(i1,k) = 0 & i1 + 1 <= Radix(k -' 1) by Def3,INT_1:7;
      hence thesis by A6,XREAL_1:19;
    end;
    case
A7:   Radix(k-'1) <= i1;
      i1 <= Radix(k) + -1 by A2,RADIX_1:13;
      then
A8:   0 - 1 <= Radix(k-'1) - 1 & i1 - Radix(k) <= -1 by XREAL_1:9,20;
      SDSub_Add_Carry(i1,k) = 1 & Radix(k-'1) + -Radix(k) <= i1 + -Radix(
      k) by A7,Def3,XREAL_1:6;
      hence thesis by A1,A8,Lm3,XXREAL_0:2;
    end;
  end;
  hence thesis;
end;
