reserve a,x,y for object, A,B for set,
  l,m,n for Nat;

theorem
  for A being finite set, X being non empty Subset-Family of A ex C
  being Element of X st for B being Element of X holds B c= C implies B = C
proof
  let A be finite set, X be non empty Subset-Family of A;
  reconsider D = COMPLEMENT X as non empty Subset-Family of A by SETFAM_1:32;
  consider x being set such that
A1: x in D and
A2: for B being set st B in D holds x c= B implies B = x by FINSET_1:6;
  reconsider x as Subset of A by A1;
  reconsider C = x` as Element of X by A1,SETFAM_1:def 7;
  take C;
  let B be Element of X such that
A3: B c= C;
  B`` = B;
  then B` in D by SETFAM_1:def 7;
  then B` = x by A2,A3,SUBSET_1:16;
  hence thesis;
end;
