I'm having trouble understanding the proof of Theorem 1 on this page:

Pauls Online Notes : Linear Algebra - Finding Inverse Matrices
I don't understand how they proved the first conditional

If A in invertible, then this means that

exists and that

.

It seems that they assumed that

, selected a solution

to write

, and then mulitplied both sides by

so that

I don't understand the logic here. It just looks like circular reasoning to me. Here is a quote from the proof:

So it seems that the equation that they are multiplying by

is:

But where does that come from? How is it that substituting

into

gives

. If I look at it this way:

=

To get the right side to be equal to zero, they must have taken

, but this isn't ANY solution. Can anyone help clarify this?