
theorem Th27: :: PCAbmax:
for R being RelStr, A being StableSet of R
 st not maximals R c= A holds not maximals R c= Lower A
proof
  let R be RelStr, A be StableSet of R such that
A1: not maximals R c= A;
   consider x being object such that
A2: x in maximals R and
A3: not x in A by A1;
  assume A4: maximals R c= Lower A;
   reconsider x9 = x as Element of R by A2;
   R is non empty by A2;
   then A5: x9 is_maximal_in [#]R by A2,Def10;
   x9 in downarrow A by A3,A2,A4,XBOOLE_0:def 3;
   then consider x99 being Element of R such that
  A6: x9 <= x99 and
  A7: x99 in A by WAYBEL_0:def 15;
      now assume x99 <> x9;
        then x9 < x99 by A6;
       hence contradiction by A2,A5,WAYBEL_4:55;
      end;
  hence contradiction by A7,A3;
end;
