I am writing a research paper, in which I am proving some properties of new convolution operation $\star$ for some transform, like linearity, associativity, commutativity,distributivity, shift invariance etc.Let us denote my transform by symbol $£$.Then, first I stated by proving linearity property:
Proposition 1:If $f(x, t), g(x, t)$ and $h(x,t)$ are any functions in $L^1(\mathbb{R^2})$ then$$(k_1f+k_2g)\star h=k_1(f\star h)+k_2(g\star h)$$
Proof:
(Then after this I write, a corollary that depends on Proposition 1)
Corollary:$$£[((k_1f+k_2g)\star h)(x, t)]=k_1£[(f\star h)(x, t)]+k_2£[(g\star h)(x, t)]$$Proof: follows by Proposition 1 and the linearity property of $£$.
Similarly, I proved commutativity, associativity, distributivity, shift invariance properties for convolution operation $\star$ and named them as proposition 2,..., proposition 5 respectively.
After some of these proposition like proposition 1, proposition 2 & proposition 4, I have written corollaries! (Like above).
Finally, I proved a theorem which shows that,$$£[((S_{k, l}f)\star g)(x,t)]=£[(f\star(S_{k, l}g))(x, t)]=\dots. $$In the proof of this theorem I used proposition 5 which states that $(S_{k, l}f)\star g=f\star(S_{k, l}g)$.
My questions
Is it good to call those properties as propositions? (Or can,I write them as lemmas?)
Can proposition or lemmas have corollaries? (Like above?, what I know is, theorems have corollaries! But, calling those properties theorems is not good)
Can I also write,the theorem in last as proposition? (However, it's proof is not that obvious)
Can we use proposition in proving some theorem? (Like above)
I am new to mathematics writing. So I need help and I am confused. Any suggestions are welcomed.
Note: Operation $\star$ was introduced in my last paper and convolution theorem for the transform $£$ is also proved in the same paper! But properties & results related to the operation ★ are not established in the same paper.