
theorem
  for F being non empty finite set,
      A being non empty FinSequence of bool F holds
   (ex X being set st X is_a_system_of_different_representatives_of A) iff
   A is Hall
proof
  let F be non empty finite set, A be non empty FinSequence of bool F;
  thus (ex X being set st X is_a_system_of_different_representatives_of A)
  implies A is Hall by Th18;
    assume
A1: A is Hall;
    then consider G being Singlification of A such that
A2: G is Hall by Th31;
    for i being Element of NAT st i in dom G holds card (G.i) = 1
    proof
      let i be Element of NAT;
      assume
A3:   i in dom G;
      dom G = dom A by Def6;
      hence thesis by A1,A3,Def8;
    end; then
    ex X being set st X is_a_system_of_different_representatives_of G
      by A2,Th17;
    hence thesis by Th28;
end;
