
theorem Th22: :: ABAC0:
for R being antisymmetric transitive RelStr, A being StableSet of R,
    z being set
 st z in Upper A & z in Lower A holds z in A
proof
 let R be antisymmetric transitive RelStr, A be StableSet of R,
     z be set such that
A1: z in Upper A and
A2: z in Lower A;
 per cases;
 suppose z in A;
   hence thesis;
 end;
 suppose A3: not z in A;
 then A4: z in uparrow A by A1,XBOOLE_0:def 3;
 A5: z in downarrow A by A2,A3,XBOOLE_0:def 3;
   reconsider y = z as Element of R by A1;
   consider x being Element of R such that
A6: x <= y and
A7: x in A by A4,WAYBEL_0:def 16;
   reconsider x9 = z as Element of R by A2;
   consider y9 being Element of R such that
A8: x9 <= y9 and
A9: y9 in A by A5,WAYBEL_0:def 15;
   x <= y9 by A6,A8,YELLOW_0:def 2;
   then x = y9 by A7,A9,Def2;
  hence z in A by A6,A7,A8,YELLOW_0:def 3;
 end;
end;
