$V=\pi^{-1}(\gamma(s_0))$ and $F(s;\gamma)=L_{s\to s_0}^{\gamma}$. Conversely, if [[(\[3.4\])]{}]{} is valid for some linear isomorphisms $F(s;\gamma)$, then a straightforward calculation shows that it converts [[(\[3.1\])]{}]{} and [[(\[3.2\])]{}]{} into identities and [[(\[3.linear\])]{}]{} holds due to the linearity of $F(s;\gamma)$.

\[Prop3.2\] Let in the vector bundle $(E,\pi,B)$ be given linear transport along paths with a representation [[(\[3.4\])]{}]{} for some vector space $V$ and linear isomorphisms $F(s;\gamma)\colon \pi^{-1}(\gamma(s))\to V,\ s\in J$. Then for a vector space ${{}\mspace{3mu}{\vphantom{V}}    _{}^{\star}\mspace{-0.8mu}V}$ there exist linear isomorphisms $
{{}\mspace{3mu}{\vphantom{\mspace{-2mu}F}}    _{}^{\star}\mspace{-0.8mu}\mspace{-2mu}F}(s;\gamma)\colon \pi^{-1}(\gamma(s))\to
    {{}\mspace{3mu}{\vphantom{V}}    _{}^{\star}\mspace{-0.8mu}V},
\ s\in J
$ for which $$\label{3.6}
L_{s\to t}^{\gamma} =
    {{}\mspace{3mu}{\vphantom{\mspace{-2mu}F}}    _{}^{\star}\mspace{-0.8mu}\mspace{-2mu}F}^{-1}(t;\gamma) \circ
    {{}\mspace{3mu}{\vphantom{\mspace{-2mu}F}}    _{}^{\star}\mspace{-0.8mu}\mspace{-2mu}F}(s;\gamma),\qquad s,t\in J.$$ iff there exists a linear isomorphism $D(\gamma)\colon V\to{{}\mspace{3mu}{\vphantom{V}}    _{}^{\star}\mspace{-0.8mu}V}$ such that $$\label{3.7}
{{}\mspace{3mu}{\vphantom{\mspace{-2mu}F}}    _{}^{\star}\mspace{-0.8mu}\mspace{-2mu}F}(s;\gamma) = D(\gamma)\circ F(s;\gamma),
                \qquad s\in J.$$

If [[(\[3.7\])]{}]{} holds, then substituting $
F(s;\gamma) = D^{-1}(\gamma)\circ {{}\mspace{3mu}{\vphantom{\mspace{-2mu}F}}    _{}^{\star}\mspace{-0.8mu}\mspace{-2mu}F}(s;\gamma)
$ into [[(\[3.4\])]{}]{}, we get [[(\[3.6\])]{}]{}. Vice versa, if [[(\[3.6\])]{}]{} is valid, then from its comparison with [[(\[3.4\])]{}]{} follows that $
D(\gamma) = {{}\mspace{3mu}{\vphantom{\mspace{-2mu}F}}    _{}^{\star}\mspace{-0.8mu}\mspace{-2mu}F}(t;\gamma)
                \circ \bigl(F(t;\gamma)\bigr)^{-1}
      = {{}\mspace{3mu}{\vphantom{\mspace{-2mu}F}}    _{}^{\star}\mspace{-0.8mu}\mspace{-2mu}F}(s;\gamma)
                \circ \bigl(F(s;\gamma)\bigr)^{-1}
$ is the required (independent of $s,t\in J$) isomorphism.

The above definition and results for linear transports along paths deal with the general case concerning arbitrary vector bundles and are therefore insensitive to the dimensionality of the 