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Suppose we have four smooth maps between smoot manifolds:

$f: M \rightarrow X$ $g: X \rightarrow N$ $h: M \rightarrow Y$ $i: Y \rightarrow N$

an the equation on compositions of jets

$j_m(g \circ f) = j_m(i \circ h)$

Then are there allways representatives f' \in j_mf g' \in j_xg h' \in j_mh i' \in j_yi

with $f(m)=x$ and $h(m)=y$ and

(g \circ f)(m') = (i \circ h)(m')

for all m' on a neighbourood of $m$ ?

I guess it is yes but I can't see how to proof it.

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    What is the point here? This is not a students guide. If you know the answer post it and don't blame me giving 'hints' ...2012-02-10

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