Proof: Rank of block of matrix is smaller than rank of matrix

Question

Let $$A = \begin{bmatrix} A1 \\ A2\end{bmatrix}$$ be a matrix with real entries

Then proof $Rank(Ai) ≤ Rank(A)$ for $i = 1, 2$

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I am attaching my solution sheet: Solution

Can someone help me understand the highlighted lines? How is "at least t pivots" guranteed?

Answer 1

Hint: if $k$ rows of $A_1$ are independent then they are also independent in $A$.