
theorem Th37:
  for A being LinearOrder holds
    the InternalRel of A linearly_orders the carrier of A
proof
  let A be LinearOrder;
  A is reflexive transitive antisymmetric connected;
  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_2:def 2, ORDERS_2:def 3, ORDERS_2:def 4;
  hence thesis by ORDERS_1:def 9;
end;
