Is |1−i||1-i| larger than |1||1|?

Let me do something very different for me. Let me give a geometric proof:

Note that $1-i$ is the bottom-right corner of the square whose center is $0$ and whose edges have length $2$. The other corners are $1+i; -1+i; -1-i$.

The diagonal running from $-1+i$ to $1-i$ is a straight line passing through $0$. Its length, by the Pythagoras theorem, is $2\sqrt2=\sqrt8$. Therefore the distance between $0$ and each of the corners is exactly half, i.e. $\sqrt2$.

And it is trivial to see that $\sqrt2>1$.

Here is a drawing:

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