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

theorem Th6:
  for C be non empty Poset, A being non empty Element of symplexes
  (C) st card A = n holds dom(SgmX(the InternalRel of C, A)) = Seg n
proof
  let C be non empty Poset, A be non empty Element of symplexes(C);
  set f = SgmX(the InternalRel of C, A);
  A in {A1 where A1 is Element of Fin the carrier of C : the InternalRel
  of C linearly_orders A1};
  then
A1: ex A1 being Element of Fin the carrier of C st A1 = A & the InternalRel
  of C linearly_orders A1;
  assume card A = n;
  then len f = n by A1,PRE_POLY:11;
  hence thesis by FINSEQ_1:def 3;
end;
