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I have spent the last 5 days stuck on this last lines of Hatcher's proof of the Van Kampen's theorem:

enter image description here

My issue is this: why is any of this needed? Isn't it already established (by the definition of our partition of $I \times I$) that each $f_i$ is mapped into the corresponding $A_{\alpha}$?

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    $\begingroup$ That's a very tiny snapshot. Hard to read. $\endgroup$ Jun 4, 2019 at 21:13
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    $\begingroup$ And very blurry. $\endgroup$
    – Lee Mosher
    Jun 4, 2019 at 21:31

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