In noncommutative *geometry* when one want to go to the differentiable level, one is forced to work with algebras which are no longer $C^*$. It is nice if we don't loose much information by the replacement of $C^*$-algebras by more general algebra. One possible choice is the so called *pre $C^*$-algebra* which is by definition a subalgebra $\mathcal{A}$ of a $C^*$-algebra $A$ of which is complete with respect to some locally convex topology finer that the topology of $A$ and is closed under the holomorphic functional calculus. Suppose that we have such $\mathcal{A}$ which is moreover a Fréchet algebra and assume that $\mathcal{A}$ is dense in $A$ (in the norm topology of $A$). The claim is that:

The inclusion $i:\mathcal{A} \to A$ induces an isomorphism in $K_0$ groups.

I found the proof of this fact in the book "Elements of Noncommutative Geometry" by Várilly, Gracia-Bondía and Figueroa. There are two moments which are not clear for me:

1. The first is that authors state that it is enough to prove this fact for *unital* algebras. I believe that it is indeed the case however I'm not sure whether there are no technical problems.

2. The second thing is the end of the proof: let us recall that $K$-theory may be defined in terms of *idempotents* (not *projections*, i.e. our idempotents need not to be self-adjoint) in matrix algebras $Q_n(A)=\{e \in M_n(A): e^2=e\}$ (the same for $\mathcal{A}$: one checks that matrix algebras for $\mathcal{A}$ are again Fréchet pre $C^*$-algebras).
Of this the quotient space is made by equivalence relation where $e \sim f$ iff there is some invertible matrix $g$ (possibly bigger that $e$ and $f$) such that $(e \oplus 0)g=g(f \oplus 0)$. At the end of the proof authors arrived to the conclusion that the inclusions $Q_n(\mathcal{A} \to Q_n(A)$ are all homotopy equivalences and they claim that this finishes the proof provided we are able to prove that two idempotents which are homotopic are equivalent in the above sense. I don't quite see two things:

Why it is enough to prove this to conclude that the inclusion induces iso on $K_0$ groups?

And also

How to prove that being homotopic implies being equivalent (in the context of general idempotents and invertibles)?