
theorem Th5:
  for S be non empty finite set,
  s be FinSequence of S holds
  (for x be set holds FDprobability(x,s)= 0)
  iff s is empty
  proof
    for S be non empty finite set,
    s be FinSequence of S holds
    (for x be set holds FDprobability(x,s)= 0) implies s is empty
    proof
      let S be non empty finite set,
      s be FinSequence of S;
      assume A1: for x be set holds FDprobability(x,s)= 0;
      assume A2: not s is empty;
      1 in dom s by A2,FINSEQ_5:6;
      then A3: s.1 in rng s by FUNCT_1:3; then
      reconsider y = s.1 as Element of S;
      A4:s"{y} <>{} by A3,FUNCT_1:72;
      FDprobability(y,s)= 0 by A1;
      hence contradiction by A4,A2;
    end;
    hence thesis;
  end;
