
theorem Th26: :: PCAamin:
for R being RelStr, A being StableSet of R
 st not minimals R c= A holds not minimals R c= Upper A
proof
  let R be RelStr, A be StableSet of R;
  assume not minimals R c= A;
  then consider x being object such that
A1: x in minimals R and
A2: not x in A;
  assume A3: minimals R c= Upper A;
   reconsider x9 = x as Element of R by A1;
   R is non empty by A1;
   then A4: x9 is_minimal_in [#]R by A1,Def9;
      x9 in uparrow A by A2,A1,A3,XBOOLE_0:def 3;
      then consider x99 being Element of R such that
  A5: x99 <= x9 and
  A6: x99 in A by WAYBEL_0:def 16;
      now assume x99 <> x9;
        then x99 < x9 by A5;
       hence contradiction by A1,A4,WAYBEL_4:56;
      end;
   hence contradiction by A6,A2;
end;
