
theorem Th7:
  for k,n be Nat st 1 in Seg n & k >= 2 holds DigA(DecSD(1,n,k),1) = 1
proof
  let k,n be Nat;
  assume that
A1: 1 in Seg n and
A2: k >= 2;
A3: Radix(k) > 2 by A2,RADIX_4:1;
  then
A4: Radix(k) > 1 by XXREAL_0:2;
  DigA(DecSD(1,n,k),1) = DigitDC(1,1,k) by A1,RADIX_1:def 9
    .= (1 mod (Radix(k) |^ 1)) div (Radix(k) |^ (1-'1)) by RADIX_1:def 8
    .= (1 div (Radix(k) |^ (1-'1))) mod Radix(k) by A3,RADIX_2:4
    .= (1 div (Radix(k) |^ 0)) mod Radix(k) by NAT_2:8
    .= (1 div 1) mod Radix(k) by NEWTON:4
    .= 1 mod Radix(k) by NAT_2:4;
  hence thesis by A4,NAT_D:14;
end;
