reserve Y for non empty set;
reserve B for Subset of Y;

theorem
  B_INF(O_el(Y)) = O_el(Y) & B_INF(I_el(Y))=I_el(Y) & B_SUP(O_el(Y)) =
  O_el(Y) & B_SUP(I_el(Y))=I_el(Y)
proof
A1: not (for x being Element of Y holds (O_el Y).x= TRUE)
  proof
    now
      assume for x being Element of Y holds (O_el Y).x= TRUE;
      let x be Element of Y;
      (O_el Y).x= FALSE by Def10;
      hence thesis;
    end;
    hence thesis;
  end;
A2: not (for x being Element of Y holds (I_el Y).x= FALSE)
  proof
    now
      assume
A3:   for x being Element of Y holds (I_el Y).x= FALSE;
      let x be Element of Y;
      (I_el Y).x= FALSE by A3;
      hence thesis by Def11;
    end;
    hence thesis;
  end;
  ( for x being Element of Y holds (O_el Y).x= FALSE)& for x being Element
  of Y holds (I_el Y).x= TRUE by Def10,Def11;
  hence thesis by A1,A2,Def13,Def14;
end;
