
theorem Th6:
  for n,k,x be Nat st n >= 1 & x needs_digits_of n,k holds DigA(
  DecSD(x,n,k),n) > 0
proof
  let n,k,x be Nat;
  assume that
A1: n >= 1 and
A2: x needs_digits_of n,k;
  x < (Radix(k) |^ n) by A2;
  then
A3: x mod (Radix(k) |^ n) = x by NAT_D:24;
  n in Seg n by A1,FINSEQ_1:3;
  then
A4: DigA(DecSD(x,n,k),n) = DigitDC(x,n,k) by RADIX_1:def 9
    .= x div (Radix(k) |^ (n -'1)) by A3,RADIX_1:def 8;
A5: (Radix(k) |^ (n-'1)) > 0 by PREPOWER:6,RADIX_2:6;
  x >= (Radix(k) |^ (n-'1)) by A2;
  hence thesis by A4,A5,NAT_2:13;
end;
