
theorem Th36: :: Pmaxmin:
for R being with_finite_clique# non empty antisymmetric transitive RelStr,
    x being Element of R
 holds ex y being Element of R st y is_minimal_in [#]R & (y = x or y < x)
proof
 let R be with_finite_clique# non empty antisymmetric transitive RelStr,
     x be Element of R;
 set sx = Lower {x}; set sL = subrelstr sx;
reconsider sL as with_finite_clique# non empty antisymmetric transitive RelStr;
   consider y being object such that
A1: y in minimals sL by XBOOLE_0:def 1;
   reconsider y as Element of sL by A1;
A2: [#]sL = sx by YELLOW_0:def 15;
then A3: y is_minimal_in sx by A1,Def9;
   reconsider y9 = y as Element of R by YELLOW_0:58;
   take y9;
   sx c= the carrier of sL by YELLOW_0:def 15;
   hence y9 is_minimal_in [#]R by A3,Th33,Th24;
   per cases;
   suppose y9 = x;
     hence thesis;
   end;
   suppose y9 <> x;
     then not y9 in {x} by TARSKI:def 1;
     then y9 in downarrow x by A2,XBOOLE_0:def 3;
     then y9 <= x by WAYBEL_0:17;
    hence thesis;
   end;
end;
