
theorem Th43: ::  WWA4a:
  for X being finite non empty set, F being Full-family of X ex G
  being Subset-Family of X st G is_generator-set_of saturated-subsets F & F = X
  deps_encl_by G
proof
  let X be finite non empty set, F be Full-family of X;
  set G = saturated-subsets F;
  take G;
A1: G is (B1) (B2) by Th32;
  thus G is_generator-set_of G
  proof
    set H = { Intersect S where S is Subset-Family of X: S c= G };
    thus G c= G;
    now
      let x be object;
      hereby
        reconsider xx=x as set by TARSKI:1;
        set sx = {xx};
        assume
A2:     x in G;
        then
A3:     sx c= G by ZFMISC_1:31;
        reconsider sx as Subset-Family of X by A2,ZFMISC_1:31;
A4:     Intersect sx = meet sx by SETFAM_1:def 9;
        Intersect sx in H by A3;
        hence x in H by A4,SETFAM_1:10;
      end;
      assume x in H;
      then
A5:   ex S being Subset-Family of X st Intersect S = x & S c= G;
      thus x in G by A1,A5,Th1;
    end;
    hence thesis by TARSKI:2;
  end;
  thus thesis by A1,Th35;
end;
