
theorem Th34: ::  WWA3_00:
  for X being finite non empty set, B being Subset-Family of X
  holds B c= saturated-subsets (X deps_encl_by B)
proof
  let X be finite non empty set, B be Subset-Family of X;
  set F = X deps_encl_by B;
  reconsider F9 = F as Full-family of X by Th33;
  set M = Maximal_wrt F9;
  let x be object;
  assume
A1: x in B;
  then reconsider x9 = x as Element of B;
  reconsider x99 = x as Subset of X by A1;
  M is (M1) by Th28;
  then consider a9, b9 being Subset of X such that
A2: [a9,b9] >= [x99,x99] and
A3: [a9, b9] in M;
A4: a9 c= x99 by A2;
  [a9,b9] in F by A3;
  then consider a, b being Subset of X such that
A5: [a9,b9] = [a,b] and
A6: for c being set st c in B & a c= c holds b c= c;
A7: a ^|^ b, F by A3,A5;
  a9 = a by A5,XTUPLE_0:1;
  then
A8: b c= x9 by A1,A4,A6;
A9: b9 = b by A5,XTUPLE_0:1;
  x99 c= b9 by A2;
  then b = x by A9,A8,XBOOLE_0:def 10;
  hence thesis by A7;
end;
