
theorem
  for n,k be Nat st n >= 1 & k >= 2 holds for tx,ty,tz be Tuple of n,k
-SD st (for i be Nat st i in Seg n holds (DigA(tx,i) = DigA(tz,i) & DigA(ty,i)
  = 0) or (DigA(ty,i) = DigA(tz,i) & DigA(tx,i) = 0)) holds SDDec(tz) + SDDec(
  DecSD(0,n,k)) = SDDec(tx) + SDDec(ty)
proof
  let n,k be Nat;
  assume
A1: n >= 1 & k >= 2;
  let tx,ty,tz be Tuple of n,k-SD;
  assume
A2: for i be Nat st i in Seg n holds DigA(tx,i) = DigA(tz,i) & DigA(ty,i
  ) = 0 or DigA(ty,i) = DigA(tz,i) & DigA(tx,i) = 0;
  for i be Nat st i in Seg n holds DigA(tx,i) = DigA(tz,i) & DigA(ty,i) =
DigA(DecSD(0,n,k),i) or DigA(ty,i) = DigA(tz,i) & DigA(tx,i) = DigA(DecSD(0,n,k
  ),i)
  proof
    let i be Nat;
    assume
A3: i in Seg n;
    then DigA(DecSD(0,n,k),i) = 0 by Th5;
    hence thesis by A2,A3;
  end;
  hence thesis by A1,Th14;
end;
