Zurückziehen von Differentialformen/Verträglichkeit mit skalarer Funktion/Aufgabe/Lösung

Es sei ${}P\in L$ und ${}v_{1},\ldots ,v_{k}\in T_{P}L$ . Dann ist

{}{\begin{aligned}\psi ^{*}{\left(f\omega )(P,v_{1},\ldots ,v_{k}\right)}&=(f\omega ){\left(\psi (P),T_{P}\psi (v_{1}),\ldots ,T_{P}\psi (v_{k})\right)}\\&=f(\psi (P))\cdot \omega {\left(\psi (P),T_{P}\psi (v_{1}),\ldots ,T_{P}\psi (v_{k})\right)}\\&={\left(\psi ^{*}(f)\right)}(P)\cdot \omega {\left(\psi (P),T_{P}\psi (v_{1}),\ldots ,T_{P}\psi (v_{k})\right)}\\&={\left(\psi ^{*}(f)\right)}(P)\cdot {\left(\psi ^{*}(\omega )\right)}{\left(P,v_{1},\ldots ,v_{k}\right)}\\&={\left(\psi ^{*}(f)\psi ^{*}(\omega )\right)}{\left(P,v_{1},\ldots ,v_{k}\right)}.\end{aligned}}