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

theorem Th3:
  2 <= k implies DigA_SDSub(SD2SDSub(DecSD(m,n,k)),n+1) =
  SDSub_Add_Carry( DigA(DecSD(m,n,k),n), k)
proof
  assume
A1: 2 <= k;
  0 + 1 <= n + 1 by XREAL_1:7;
  then
A2: (n+1) in Seg (n+1) by FINSEQ_1:1;
  hence
  DigA_SDSub(SD2SDSub(DecSD(m,n,k)),n+1) = SD2SDSubDigitS(DecSD(m,n,k), n
  +1, k) by RADIX_3:def 8
    .= SD2SDSubDigit(DecSD(m,n,k), n+1, k) by A1,A2,RADIX_3:def 7
    .= SDSub_Add_Carry( DigA(DecSD(m,n,k),n+1-'1), k) by RADIX_3:def 6
    .= SDSub_Add_Carry( DigA(DecSD(m,n,k),n), k) by NAT_D:34;
end;
