Invariance of Pseudoinverse under Addition of Degenerate Transformation

Theorem
Let $U, V$ be vector spaces over a field $K$.

Let $S: U \to V$ be a linear transformation.

Let $T: V \to U$ be a linear transformation.

Let $S$ and $T$ are pseudoinverse to each other.

Then $S + G_1$ and $T + G_2$ are pseudoinverse to each other, where:
 * $G_1: U \to V$ is an arbitrary degenerate linear transformation
 * $G_2: V \to U$ is an arbitrary degenerate linear transformation

Proof
Let:

By, $G_3, G_4$ are degenerate.

Then:

which is degenerate in view of:
 * Product with Degenerate Linear Transformation is Degenerate
 * Right Product with Degenerate Linear Transformation is Degenerate
 * Product with Degenerate Linear Transformation is Degenerate

Similarly:

which is degenerate.