
theorem FieldGen1:
  for X being set, P being semialgebra_of_sets of X holds
    P c= Field_generated_by P
proof
   let X be set, P be semialgebra_of_sets of X;
   set Y = {Z where Z is Field_Subset of X : P c= Z};
A1:bool X in Y;
   for A being set st A in Y holds P c= A
   proof
    let A be set;
    assume A in Y; then
    ex Z being Field_Subset of X st A = Z & P c= Z;
    hence P c= A;
   end;
   hence thesis by A1,SETFAM_1:5;
end;
