
theorem
  for X being set, B being non empty finite Subset-Family of X st B is
  (B1) (B2) holds /\-IRR B is_generator-set_of B
proof
  let X be set, B be non empty finite Subset-Family of X;
  assume
A1: B is (B1) (B2);
  set G = /\-IRR B;
  set H = {Intersect S where S is Subset-Family of X:S c= G};
  thus G c= B;
  now
    let x be object;
    hereby
      assume x in B;
      then reconsider xx = x as Element of B;
      consider yz being non empty Subset of B such that
A2:   xx = meet yz and
A3:   for s being set st s in yz holds s is_/\-irreducible_in B by Th3;
      reconsider yz as non empty Subset-Family of X by XBOOLE_1:1;
A4:   yz c= G
      proof
        let x be object;
         reconsider xx=x as set by TARSKI:1;
        assume x in yz;
        then xx is_/\-irreducible_in B by A3;
        hence thesis by Def3;
      end;
      Intersect yz = meet yz by SETFAM_1:def 9;
      hence x in H by A2,A4;
    end;
    assume x in H;
    then ex S being Subset-Family of X st x = Intersect S & S c= G;
    hence x in B by A1,Th1,XBOOLE_1:1;
  end;
  hence thesis by TARSKI:2;
end;
