
theorem
  for m,k,f be Nat, r be Tuple of (m+2),k-SD st m >= 1 & k >= 2 & f
needs_digits_of m,k holds ex s be Integer st - f < (SDDec(M0(r)) - s*f) & SDDec
  (Mmax(r)) - s*f < f
proof
  let m,k,f be Nat, r be Tuple of (m+2),k-SD;
  assume that
A1: m >= 1 and
A2: k >= 2 and
A3: f needs_digits_of m,k;
  SDDec(Fmin(m+2,m,k)) <= f by A1,A2,A3,Th7;
  then
A4: SDDec(M0(r)) + SDDec(Fmin(m+2,m,k)) <= SDDec(M0(r)) + f by XREAL_1:7;
  SDDec(Mmax(r)) < SDDec(M0(r)) + SDDec(Fmin(m+2,m,k)) by A1,A2,Th10;
  then
A5: SDDec(Mmax(r)) < SDDec(M0(r)) + f by A4,XXREAL_0:2;
  Radix(k) |^ (m-'1) > 0 by NEWTON:83,RADIX_2:6;
  then f > 0 by A3;
  hence thesis by A5,Th8;
end;
