
theorem
  for X being set, F being Field_Subset of X
    ex S being semialgebra_of_sets of X
    st F = S & F = Field_generated_by S
proof
   let X be set, F be Field_Subset of X;
   reconsider S = F as semialgebra_of_sets of X by SRINGS_3:20;
   take S;
   now let x be object;
    assume x in Field_generated_by S; then
A2: x in meet{Z where Z is Field_Subset of X : S c= Z} by SRINGS_3:def 7;
    F in {Z where Z is Field_Subset of X: S c= Z};
    hence x in S by A2,SETFAM_1:def 1;
   end; then
   Field_generated_by S c= S;
   hence thesis by SRINGS_3:21;
end;
