reserve n,m,k for Nat,
  x,y for set,
  r for Real;
reserve C,D for non empty finite set,
  a for FinSequence of bool D;

theorem Th9:
  for a be terms've_same_card_as_number length_equal_card_of_set
  FinSequence of bool D ex d be Element of D st a.1 = {d}
proof
  let A be terms've_same_card_as_number length_equal_card_of_set FinSequence
  of bool D;
  len A <> 0 by Th4;
  then
A1: 0+1<=len A by NAT_1:13;
  then reconsider A1 = A.1 as finite set by Lm2;
  card(A1)=1 by A1,Def1;
  then consider x being object such that
A2: {x} = A.1 by CARD_2:42;
  1 in dom A by A1,FINSEQ_3:25;
  then A.1 c= D by Lm5;
  then reconsider x as Element of D by A2,ZFMISC_1:31;
  take x;
  thus thesis by A2;
end;
