
theorem Th38: :: Pminmax
for R being with_finite_clique# non empty antisymmetric transitive RelStr,
    x being Element of R
  ex y being Element of R st y is_maximal_in [#]R & (y = x or x < y)
proof
 let R be with_finite_clique# non empty antisymmetric transitive RelStr,
     x be Element of R;
 set ax = Upper {x}; set sU = subrelstr ax;
reconsider sU as with_finite_clique# non empty antisymmetric transitive RelStr;
 consider y being object such that
A1: y in maximals sU by XBOOLE_0:def 1;
 reconsider y as Element of sU by A1;
A2: [#]sU = ax by YELLOW_0:def 15;
then A3: y is_maximal_in ax by A1,Def10;
 reconsider y9 = y as Element of R by YELLOW_0:58;
   take y9;
   ax c= the carrier of sU by YELLOW_0:def 15;
   hence y9 is_maximal_in [#]R by A3,Th32,Th25;
   per cases;
   suppose y9 = x;
     hence thesis;
   end;
   suppose y9 <> x;
     then not y9 in {x} by TARSKI:def 1;
     then y9 in uparrow x by A2,XBOOLE_0:def 3;
     then x <= y9 by WAYBEL_0:18;
    hence thesis;
   end;
end;
