
theorem
for X be set, S be with_empty_element semi-diff-closed
     cap-closed Subset-Family of X holds
  sigmaring(Ring_generated_by S) = sigmaring S
proof
   let X be set, S be with_empty_element semi-diff-closed
      cap-closed Subset-Family of X;
A1:S c= Ring_generated_by S by RingGen1;
   Ring_generated_by S c= sigmaring(Ring_generated_by S) by Defsigmaring; then
A2:S c= sigmaring(Ring_generated_by S) by A1;
   now let x be object;
    assume A3: x in Ring_generated_by S;
    S c= sigmaring S by Defsigmaring; then
    sigmaring S in {Z where Z is non empty preBoolean Subset-Family of X
                     : S c= Z};
    hence x in sigmaring S by A3,SETFAM_1:def 1;
   end; then
   Ring_generated_by S c= sigmaring S;
   hence thesis by A2,Defsigmaring;
end;
