Index of Square Triangular Number from Preceding

Theorem
Let $T_n$ be the $n$th triangular number.

Let $T_n$ be square.

Then $T_{4 n \paren {n + 1} }$ is also square.

Proof
The product of two squares is also square.

Let $T_n$ be square.

Therefore $T_{4 n \paren {n + 1} }$ is also square.