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

theorem Th5:
  for k,x,n be Nat st n >= 1 & k >= 3 & x is_represented_by (n+1),k
  holds DigA_SDSub(SD2SDSub(DecSD(x mod (Radix(k) |^ n),n,k)),n+1) =
  SDSub_Add_Carry(DigA(DecSD(x,n,k),n),k)
proof
  let k,x,n be Nat;
  assume that
A1: n >= 1 and
A2: k >= 3;
  set xn = x mod (Radix(k) |^ n);
A3: n+1 in Seg (n+1) by FINSEQ_1:3;
  then
  DigA_SDSub(SD2SDSub(DecSD(xn,n,k)),n+1) = SD2SDSubDigitS(DecSD(xn,n,k),n
  +1,k) by RADIX_3:def 8
    .= SD2SDSubDigit(DecSD(xn,n,k),n+1,k) by A2,A3,RADIX_3:def 7,XXREAL_0:2
    .= SDSub_Add_Carry(DigA(DecSD(xn,n,k),n+1-'1),k) by RADIX_3:def 6
    .= SDSub_Add_Carry(DigA(DecSD(xn,n,k),n+0),k) by NAT_D:34
    .= SDSub_Add_Carry(DigA(DecSD(x,n,k),n),k) by A1,Lm3;
  hence thesis;
end;
