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

theorem
  {} in symplexes(C)
proof
 {} is Subset of C by SUBSET_1:1;
  then reconsider a = {} as Element of Fin the carrier of C by FINSUB_1:def 5;
A1: the InternalRel of C is_antisymmetric_in a;
A2: the InternalRel of C is_connected_in a;
A3: the InternalRel of C is_transitive_in a;
  the InternalRel of C is_reflexive_in a;
  then the InternalRel of C linearly_orders a by A1,A3,A2,ORDERS_1:def 9;
  hence thesis;
end;
