
theorem
  for X, G being set, B being non empty finite Subset-Family of X st B
  is (B1) (B2) & G is_generator-set_of B holds /\-IRR B c= G\/{X}
proof
  let X, G be set, B be non empty finite Subset-Family of X such that
A1: B is (B1) (B2) and
A2: G is_generator-set_of B;
A3: B = { Intersect S where S is Subset-Family of X: S c= G } by A2;
A4: G c= B by A2;
  let x be object;
  assume
A5: x in /\-IRR B;
  then reconsider xx = x as Element of B;
A6: xx is_/\-irreducible_in B by A5,Def3;
  assume
A7: not x in G\/{X};
  then not x in {X} by XBOOLE_0:def 3;
  then
A8: x <> X by TARSKI:def 1;
  x in B by A5;
  then consider S being Subset-Family of X such that
A9: x = Intersect S and
A10: S c= G by A3;
  not x in S by A10,A7,XBOOLE_0:def 3;
  hence contradiction by A1,A4,A9,A10,A8,A6,Th4,XBOOLE_1:1;
end;
