Yes, the sentence is correct.
To take part in - you join a group, institution (what you have), usually by afinitity. That is, because you have a common purpose, ideology or interest.
To take part with - you are doing some activity with someone (taking part). The activity [*] or event will be expressed.
"To take part with" most of the time, it means physically.
One example to illustrate the point (I mean, with the usage of "with"):
Review the information below and answer the following question.
For a panel of professors to asses the State of the Union Message on public TV, the producer must choose two Republicans and two Democrats. At least one professor must be an economist and at least one military expert.
Available Republican's are Alan, Brown, Caldwell, Darvey, and Elzer.
Available Democrat's are Francis, Garland, Humes, and Indroff.
Caldwell, Francis, and Garland are economist.
Darvey and Indroff are military experts.
Francis will not sit in the same room with Caldwell, and will take part only if Alan is on the panel.
Darvey refuses to take part with Garland,
Elzer refuses to take part with Indroff.
If Garland is chosen, which of the following must be true?
I. Any acceptable panel must contain Indroff.
II. Any acceptable panel must contain Alan.
III. There is no acceptable panel which contains Brown
A) I only
B) I, II, and III
C) II only
D) I and II only
E) II and III only
We can see from the question that it´s not that Darvey doesn´t want to to join the panel, what he refuses is to be in the panel (assessing the State of the Union) with Garland.