Complex sentence: In the current iteration, after inverting the path, d will be free on u, and by the previous result, it will also be free on w. Rotating F' does not compromises the validity of the coloring.
Split into simpler sentences: In the current iteration, after inverting the path, "d" will be free on "u", and by the previous result, it will also be free on "w". Rotating "F"' does not compromises the validity of the coloring.