(Ugarte, Probability and Statistics with R, 2E) In the book, it reads:
$SST$: Sum of squares of Total.
$SSE$: Sum of squares of Error.
$SSR$: Sum of squares of Regression.
Marginal increases when adding x2 to a model that already contains x1 will be denoted as
$SSR(x2|x1) = SSR(x2,x1) - SSR(x1)$
which is equivalent to
$SSR(x2|x1) = SSE(x1) - SSE(x1,x2)$.
Why is there an equivalence?
What I did:
$SSR(x2|x1) = SSR(x2,x1) - SSR(x1)$ (using $SST=SSR+SSE$)
$= SST(x2,x1) - SSE(x2,x1) - [SST(x1) - SSE(x1)] $
$= SSE(x1) - SSE(x2,x1) + [SST(x2,x1) - SST(x1)]$.
So, in fact, the question turns out to be why $SST(x2,x1) - SST(x1) = 0$!