The principal symbol depends only on the first derivative of a smooth coordinate change

For a first $y$-derivative, [L2] gives ${\mathrm{\partial }}_{y_j}v(y)={\sum }_{i=1}^n{\mathrm{\partial }}_{x_i}u(\mathrm{\Phi }(y)){\mathrm{\partial }}_{y_j}{\mathrm{\Phi }}_i(y)$, so each differentiation either lands on a derivative of $u$ or on one factor ${\mathrm{\partial }}_{y_j}{\mathrm{\Phi }}_i$.

Repeating step 1.1 $m$ times produces a sum of terms, and the only terms containing an order-$m$ derivative of $u$ are those in which every differentiation lands on the $u$-factor; once a differentiation lands on a coefficient ${\mathrm{\partial }}_{y_j}{\mathrm{\Phi }}_i$, the remaining differentiations can raise the order of the $u$-derivative by at most $m-1$, so every derivative of $\mathrm{\Phi }$ of order at least $2$ contributes only to lower-order terms of $\tilde{L}$.

In the top-order part, each application of step 1.1 contributes one factor of the pulled-back covector, and [L3] lets us reorder the $m$ differentiations without changing the result; therefore [L1] identifies the transformed principal symbol as ${\tilde{p}}_m(y,\eta )=p_m(\mathrm{\Phi }(y),D\mathrm{\Phi }(y{)}^{-T}\eta )$, which depends only on the first derivative of the coordinate change.