In this document https://courses.engr.illinois.edu/cs598csc/sp2010/Lectures/Lecture9.pdf they prove the integrality of the matching polytope using the integrality of the perfect matching polytope.
The only part I don't understand is on page 4-5, when they claim that $\tilde x(\tilde \delta(U)) \geq \tilde x(\tilde \delta(X' \setminus W')) + \tilde x(\tilde \delta(W \setminus X))$. I don't understand why this doesn't hold with equality, since the values on copied edges in the duplicated graph are identical.
Can someone clarify this proof? Is it a mistake?