
theorem Th15:
  for F being finite set, A being non empty FinSequence of bool F
  st A is Hall holds A is non-empty
proof
  let F be finite set, A be non empty FinSequence of bool F;
  assume
A1: A is Hall;
  assume A is non non-empty;
  then {} in rng A by RELAT_1:def 9;
  then consider i being object such that
A2: i in dom A & A.i = {} by FUNCT_1:def 3;
  set J = {i};
A3: card J = 1 by CARD_2:42;
  J c= dom A & card (union (A, J)) = 0 by A2,Th5,CARD_1:27,ZFMISC_1:31;
  hence thesis by A1,A3;
end;
