Mean Curvature Flow: Geometric Invariance under Tangential Perturbations

Question

I have a problem with the proof of a theorem that is often referred to in the literature as "Geometric Invariance under Tangential Perturbations".

Context

Let us now provide a second possible definition of a mean curvature flow:

Now we want to prove the substantial equivalence of the two definitions. The theorem is the following:

I don't understand the highlighted part. How is that highlighted formula obtained? And once obtained, is (4) invoked for uniqueness?