
theorem :: Pbm:
for R being with_finite_clique# antisymmetric transitive RelStr
 holds Lower maximals R = [#]R
proof
 let P being with_finite_clique# antisymmetric transitive RelStr;
 set ap = Lower maximals P; set cP = the carrier of P;
   cP c= ap proof
     let x be object;
     assume A1: x in cP;
     then reconsider x9 = x as Element of P;
   A2: P is non empty by A1;
     then consider y being Element of P such that
   A3: y is_maximal_in [#]P and
   A4: y = x9 or y > x9 by Th38;
   A5: y in maximals P by A3,A2,Def10;
       per cases by A4;
       suppose x9 = y;
         hence thesis by A5,XBOOLE_0:def 3;
       end;
       suppose y > x9;
         then x9 <= y;
         then x in downarrow maximals P by A5,WAYBEL_0:def 15;
         hence x in ap by XBOOLE_0:def 3;
       end;
   end;
  hence thesis;
end;
