
theorem Th1:
  for n being Nat, E being non empty set st card n c=
  card E holds the_subsets_of_card(n, E) is non empty
proof
  let n be Nat;
  let E be non empty set;
  reconsider n9=n as Element of NAT by ORDINAL1:def 12;
  assume card n c= card E;
  then consider f be Function such that
A1: f is one-to-one and
A2: dom f = n and
A3: rng f c= E by CARD_1:10;
  set X = f .: n;
A4: rng f = X by A2,RELAT_1:113;
  then n,X are_equipotent by A1,A2;
  then card X = n9 by CARD_1:def 2;
  then X in {X9 where X9 is Subset of E: card X9 = n} by A3,A4;
  hence thesis;
end;
