Algebraic Topology - Retractions and Homomorpisms Induced by Inclusions

I am reading Munkres book on Topology, Part II - Algegraic Topology Chapter 9 on the Fundamental Group.

On page 348 Munkres gives the following Lemma concerned with the homomorphism of fundamental groups induced by inclusions":

" Lemma 55.1. If A is a retract of X, then the homomorphism of fundamental groups induced by inclusion $j: A \rightarrow X$ is injective"

I am struggling with the proof - not so much intuitively - but in formulating a formal and explicit proof.

Because explaining my postion requires diagrams I have set out my problem in an attachment - see the attachment "Retractions and Induced Homomorphisms.

I have also provided an attachement of the relevant pages of Munkres book

I would like as much as anything a confirmation that my reasoning in the attachment "Problem ... ... " is correct. I would also be most interested to see how to formulate a formal and explicit proof of the Lemma

Peter
Attached Files
 Problem - Retractions and Induced Homomorphisms.pdf (141.2 KB, 3 views) Munkres Pages 333 and 348 - Fundamental Group.pdf (84.8 KB, 3 views)

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