
theorem Th39: ::  Ex3:
  for X being set, K being Subset of X, B being Subset-Family of X
  st B = {X}\/{A where A is Subset of X : not K c= A} holds B is (B1) (B2)
proof
  let X be set, K be Subset of X, BB be Subset-Family of X such that
A1: BB = {X}\/{B where B is Subset of X : not K c= B};
  X in {X} by TARSKI:def 1;
  then X in BB by A1,XBOOLE_0:def 3;
  hence BB is (B1);
  set BB1 = {B where B is Subset of X : not K c= B};
  thus BB is (B2)
  proof
    let a, b be set;
    assume that
A2: a in BB and
A3: b in BB;
    reconsider a9 =a, b9 = b as Subset of X by A2,A3;
    per cases by A1,A2,A3,XBOOLE_0:def 3;
    suppose
A4:   a in {X} & b in {X};
      then
A5:   b = X by TARSKI:def 1;
      a = X by A4,TARSKI:def 1;
      then a/\ b in {X} by A5,TARSKI:def 1;
      hence thesis by A1,XBOOLE_0:def 3;
    end;
    suppose
A6:   a in {X} & b in BB1;
      then a = X by TARSKI:def 1;
      then a9/\b9 = b by XBOOLE_1:28;
      hence thesis by A1,A6,XBOOLE_0:def 3;
    end;
    suppose
A7:   a in BB1 & b in {X};
      then b = X by TARSKI:def 1;
      then a9/\b9 = a by XBOOLE_1:28;
      hence thesis by A1,A7,XBOOLE_0:def 3;
    end;
    suppose
      a in BB1 & b in BB1;
      then ex B1 being Subset of X st b = B1 & not K c= B1;
      then not K c= a/\b by XBOOLE_1:18;
      then a9/\b9 in BB1;
      hence thesis by A1,XBOOLE_0:def 3;
    end;
  end;
end;
