
theorem :: WWA4b:
  for X be finite non empty set, G being Subset-Family of X holds G
  is_generator-set_of saturated-subsets (X deps_encl_by G)
proof
  let X be finite non empty set, G be Subset-Family of X;
  set F = X deps_encl_by G;
  set ssF = saturated-subsets F;
  F is full_family by Th33;
  then
A1: ssF is (B1) (B2) by Th32;
  thus G is_generator-set_of ssF
  proof
    set H = { Intersect S where S is Subset-Family of X: S c= G };
    thus
A2: G c= ssF by Th34;
    now
      let x be object;
      hereby
        assume x in ssF;
        then consider b9, a9 being Subset of X such that
A3:     x = b9 and
A4:     a9 ^|^ b9, F by Th31;
        [a9, b9] in Maximal_wrt F by A4;
        then [a9,b9] in F;
        then consider a, b being Subset of X such that
A5:     [a, b] = [a9, b9] and
A6:     for c being set st c in G & a c= c holds b c= c;
        set C = {c where c is Subset of X: c in G & a c= c};
        C c= bool X
        proof
          let x be object;
          assume x in C;
          then ex c being Subset of X st x = c & c in G & a c= c;
          hence thesis;
        end;
        then reconsider C as Subset-Family of X;
        now
          let z1 be set;
          assume z1 in C;
          then ex c being Subset of X st z1 = c & c in G & a c= c;
          hence b c= z1 by A6;
        end;
        then
A7:     b c= Intersect C by MSSUBFAM:4;
        set ic = Intersect C;
A8:     b = b9 by A5,XTUPLE_0:1;
A9:     C c= G
        proof
          let c be object;
          assume c in C;
          then ex cc being Subset of X st cc = c & cc in G & a c= cc;
          hence thesis;
        end;
        thus x in H
        proof
          per cases;
          suppose
            b = ic;
            hence thesis by A3,A8,A9;
          end;
          suppose
A10:        b <> ic;
            reconsider ic as Subset of X;
            now
              let c be set;
              assume that
A11:          c in G and
A12:          a c= c;
              c in C by A11,A12;
              hence ic c= c by MSSUBFAM:2;
            end;
            then
A13:        [a,ic] in F;
            [a,b] <= [a,ic] by A7;
            hence thesis by A4,A5,A8,A10,A13,Th27;
          end;
        end;
      end;
      assume x in H;
      then
A14:  ex S being Subset-Family of X st Intersect S = x & S c= G;
      thus x in ssF by A1,A2,A14,Th1,XBOOLE_1:1;
    end;
    hence thesis by TARSKI:2;
  end;
end;
