
theorem Th8:
  for i,k,n be Nat st i in Seg n & i > 1 & k >= 2 holds DigA(DecSD(
  1,n,k),i) = 0
proof
  let i,k,n be Nat;
  assume that
A1: i in Seg n and
A2: i > 1 and
A3: k >= 2;
  i-1 > 1 - 1 by A2,XREAL_1:14;
  then
A4: i-'1 > 0 by XREAL_0:def 2;
A5: Radix(k) > 2 by A3,RADIX_4:1;
  then Radix(k) > 1 by XXREAL_0:2;
  then
A6: 1 div (Radix(k) |^ (i-'1)) = 0 by A4,NAT_D:27,PEPIN:25;
  DigA(DecSD(1,n,k),i) = DigitDC(1,i,k) by A1,RADIX_1:def 9
    .=(1 mod (Radix(k) |^ i)) div (Radix(k) |^ (i-'1)) by RADIX_1:def 8
    .=(1 div (Radix(k) |^ (i-'1))) mod Radix(k) by A2,A5,RADIX_2:4;
  hence thesis by A6,NAT_D:26;
end;
