What is the (simple) reason that Kirchhoff’s law is preserved in augmenting a flow by the use of a flow augmenting path?

Chapter 23, PROBLEM SET 23.7 #4

What is the (simple) reason that Kirchhoff’s law is preserved in augmenting a flow by the use of a flow augmenting path?

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Related Book For answer-question

Advanced Engineering Mathematics

10th edition

Authors: Erwin Kreyszig

ISBN: 978-0470458365