Another time on jets and composition

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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migrated from meta.math.stackexchange.comFeb 8 '12 at 6:32

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Hint for your actual question: you are looking at a purely local statement, so you can replace all your smooth manifolds by neighbour hoods of 0 in Euclidean spaces, and assume that $m = x = y = g(x) = i(y) = 0$. So your statement becomes one about Taylor polynomials. Can you see how to solve the problem in this case? – Willie Wong Feb 8 '12 at 9:25
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' ... – Mark Neuhaus Feb 10 '12 at 8:33