What's it called when, in a proof, we define a new variable/function in terms of two existing ones, in order to make it easier to write or follow?

Question

Say, for instance, we're trying to prove a theorem involving two defined functions $f$ and $g$, and $x \in \mathbb{R}$.

Within this proof, we have to deal with the sum $f(x) + g(x)$ multiple times.

So we decide, at some point, to write:

$$ \text{call } h(x) = f(x) + g(x)$$

to make the proof easier to write or read.

1. What do we call this operation?

2. And could we write 'let' instead of 'call'?

Answer 1

1. I don't think this operation has any name. I would describe the step as "introducing" or "constructing" a new function
2. I think "let" is acceptable, but (and this opinion is purely subjective) not quite as good. "let" is usually there to instantiate universally quantified variables, that is, if the statement starts with $\forall \epsilon > 0$, we start the proof by saying "Let $\epsilon>0$" to say that we allow $\epsilon$ to be any positive number. Instead of either call or let, I would say "define" or "construct".

Answer 2

That proof strategy does not have a name.

"Let" is much better than "call". You want your reader to know that you are defining something, not simply renaming something they might think they should remember from earlier.

Edit: @5xum suggests "define" instead of "let". That is probably better.