
theorem
  for X being non empty set, S being semialgebra_of_sets of X holds
    sigma (Field_generated_by S) = sigma S
proof
   let X be non empty set, S be semialgebra_of_sets of X;
A1:S c= Field_generated_by S by FieldGen1;
   Field_generated_by S c= sigma(Field_generated_by S) by PROB_1:def 9; then
A2:S c= sigma(Field_generated_by S) by A1;
   now let x be object;
    assume A3: x in Field_generated_by S;
    S c= sigma S by PROB_1:def 9; then
    sigma S in {Z where Z is Field_Subset of X : S c= Z};
    hence x in sigma S by A3,SETFAM_1:def 1;
   end; then
   Field_generated_by S c= sigma S;
   hence thesis by A2,PROB_1:def 9;
end;
