Query using subset of path in Postgres

Given this table:

 id |            points (path)                 |
----+------------------------------------------+
  1 | ((1,2),(3,4),(5,6),(7,8))                |

Is it possible to achieve the following using a single geometric operator and a path argument (sequential subset of the containing path), like ((3,4),(5,6))?

select * from things where points @> '(3,4)' and points @> '(5,6)';