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

theorem Th2:
  for x,y be Integer, k be Nat st 3 <= k holds SDSub_Add_Carry(
  SDSub_Add_Carry(x,k) + SDSub_Add_Carry(y,k), k ) = 0
proof
  let x,y be Integer, k be Nat;
  set CC = SDSub_Add_Carry(x,k) + SDSub_Add_Carry(y,k);
  -1 <= SDSub_Add_Carry(x,k) & -1 <= SDSub_Add_Carry(y,k) by RADIX_3:12;
  then
A1: -1 + -1 <= CC by XREAL_1:7;
  assume k >= 3;
  then
A2: k - 1 >= 3 - 1 by XREAL_1:13;
  then k - 1 > 0 by XXREAL_0:2;
  then k - 1 = k -' 1 by XREAL_0:def 2;
  then
A3: Radix(k-'1) > 2 by A2,Th1;
  SDSub_Add_Carry(x,k) <= 1 & SDSub_Add_Carry(y,k) <= 1 by RADIX_3:12;
  then CC <= 1 + 1 by XREAL_1:7;
  then
A4: CC < Radix(k-'1) by A3,XXREAL_0:2;
  -Radix(k-'1) <= -2 by A3,XREAL_1:24;
  then -Radix(k-'1) <= CC by A1,XXREAL_0:2;
  hence thesis by A4,RADIX_3:def 3;
end;
