Suppose we have two decreasing sequences $A_i=\langle a_1,..,a_i\rangle $, $B_j= \langle b_1,..,b_j\rangle $, now the decreasing union sequence of those would be the decreasing sequence $C_{i+j}=\langle c_1,..,c_{i+j}\rangle $ such that $\forall k[c_k \in rng (C_{i+j}) \leftrightarrow c_k \in rng(A_i) \lor c_k \in rng(B_j)]$, let's denote that as $C_{i+j}= A_i \cup_\downarrow B_j$. Now lets define $$ \Sigma \ [A_i] = a_1 + a_2 +...+a_i$$
Now for every ordinal $\alpha$ let $C(\alpha)$ be the decreasing sequence of ordinals in the Cantor normal form of $\alpha$.
Define: $\alpha +^\uparrow \beta = \Sigma \ [C(\alpha) \cup_\downarrow C(\beta)] $
Now I'm not sure if this is the same as Hessenberg natural sum of ordinals, that's why I gave it a different denotation here.
Now the question is: Does the following always follow in ZF?
$\gamma> \delta \to \gamma +^\uparrow \alpha > \delta +^\uparrow \alpha$