
theorem :: Pam:
for R being with_finite_clique# antisymmetric transitive RelStr
 holds Upper minimals R = [#]R
proof
 let R being with_finite_clique# antisymmetric transitive RelStr;
 set ap = Upper minimals R; set cR = the carrier of R;
   cR c= ap proof
     let x be object;
     assume A1: x in cR;
     then reconsider x9 = x as Element of R;
   A2: R is non empty by A1;
     then consider y being Element of R such that
   A3: y is_minimal_in [#]R and
   A4: y = x9 or y < x9 by Th36;
   A5: y in minimals R by A3,A2,Def9;
       per cases by A4;
       suppose x9 = y;
         hence thesis by A5,XBOOLE_0:def 3;
       end;
       suppose y < x9;
         then y <= x9;
         then x in uparrow minimals R by A5,WAYBEL_0:def 16;
         hence x in ap by XBOOLE_0:def 3;
       end;
   end;
  hence thesis;
end;
