reserve A,x,y,z,u for set,
  m,n for Element of NAT;
reserve C for non empty Poset;

theorem Th5:
  for x, s be set st x c= s & s in symplexes(C) holds x in symplexes(C)
proof
  let x, s be set;
  assume that
A1: x c= s and
A2: s in symplexes(C);
  consider s1 be Element of Fin the carrier of C such that
A3: s1 = s and
A4: the InternalRel of C linearly_orders s1 by A2;
  s1 c= the carrier of C by FINSUB_1:def 5;
  then x c= the carrier of C by A1,A3;
  then reconsider x1 = x as Element of Fin the carrier of C by A1,A2,
FINSUB_1:def 5;
  the InternalRel of C linearly_orders x by A1,A3,A4,ORDERS_1:38;
  then
  x1 in {A where A is Element of Fin the carrier of C : the InternalRel of
  C linearly_orders A};
  hence thesis;
end;
