
theorem Th32: ::  WWA3:
  for X being finite non empty set, F being Full-family of X holds
  saturated-subsets F is (B1) (B2)
proof
  let X be finite non empty set, F be Full-family of X;
  set ss = saturated-subsets F;
A1: Maximal_wrt F is (M1) by Th28;
  then consider A9, B9 being Subset of X such that
A2: [A9,B9] >= [[#]X,[#]X] and
A3: [A9,B9] in Maximal_wrt F;
  [#]X c= B9 by A2;
  then
A4: [#]X = B9 by XBOOLE_0:def 10;
  A9 ^|^ B9, F by A3;
  then X in ss by A4;
  hence ss is (B1);
  thus ss is (B2)
  proof
    let a, b be set such that
A5: a in ss and
A6: b in ss;
    reconsider a9 = a, b9 = b as Subset of X by A5,A6;
A7: Maximal_wrt F is (M3) by Th28;
    consider B2, A2 being Subset of X such that
A8: b = B2 and
A9: A2 ^|^ B2, F by A6,Th31;
A10: [A2,B2] in Maximal_wrt F by A9;
    consider B1, A1 being Subset of X such that
A11: a = B1 and
A12: A1 ^|^ B1, F by A5,Th31;
A13: [A1,B1] in Maximal_wrt F by A12;
    consider A9, B9 being Subset of X such that
A14: [A9,B9] >= [a9/\b9,a9/\b9] and
A15: [A9,B9] in Maximal_wrt F by A1;
A16: A9 c= a/\b by A14;
    a/\b c= b by XBOOLE_1:17;
    then A9 c= B2 by A8,A16;
    then
A17: B9 c= B2 by A10,A15,A7;
A18: a/\b c= B9 by A14;
    a/\b c= a by XBOOLE_1:17;
    then A9 c= B1 by A11,A16;
    then B9 c= B1 by A13,A15,A7;
    then B9 c= a/\b by A11,A8,A17,XBOOLE_1:19;
    then
A19: B9 = a/\b by A18,XBOOLE_0:def 10;
    A9 ^|^ B9, F by A15;
    hence thesis by A19;
  end;
end;
