
theorem RingGen1:
  for X being set, P being with_empty_element semi-diff-closed
    cap-closed Subset-Family of X
  holds P c= Ring_generated_by P
proof
   let X be set, P be with_empty_element semi-diff-closed
     cap-closed Subset-Family of X;
   set Y = {Z where Z is non empty preBoolean Subset-Family 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 non empty preBoolean Subset-Family of X st A = Z & P c= Z;
    hence P c= A;
   end;
   hence thesis by A1,SETFAM_1:5;
end;
