I want to prove the next assertion in Jeffrey M. Lee's Manifolds and differential geometry.
If $\mathcal{D}_1, \mathcal{D}_2$ are (natural) graded derivations of degrees $r_1,r_2$ respectively, then the operator:
$[\mathcal{D}_1,\mathcal{D}_2] := \mathcal{D}_1 \circ \mathcal{D}_2 - (-1)^{r_1 r_2} \mathcal{D}_2 \circ \mathcal{D}_1$

is a natural graded derivation of degree $r_1+r_2$.
I am finding it difficult to prove property 2 and 3 of graded derivation for this bracket.
Property 2 is given in the next page in definition 1.

I am uploading scans of my work (hopefully my hand written work won't stir you away).
Attached Thumbnails

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