
theorem
  for A being RelStr st the InternalRel of A linearly_orders the carrier of A
  holds A is reflexive transitive antisymmetric connected
proof
  let A be RelStr;
  assume the InternalRel of A linearly_orders the carrier of A;
  then the InternalRel of A is_reflexive_in the carrier of A &
    the InternalRel of A is_transitive_in the carrier of A &
    the InternalRel of A is_antisymmetric_in the carrier of A &
    the InternalRel of A is_connected_in the carrier of A
    by ORDERS_1:def 9;
  hence thesis by ORDERS_2:def 2, ORDERS_2:def 3, ORDERS_2:def 4;
end;
