electromagnetism - Transmission Line, TEM wave, transverse Laplacian operator (electromagnetic)

Time-Harmonic Electromagnetic Fields (1961) by R. F. Harrington, ISBN 07-026745-6, p. 63, he discusses TEM mode in a transmission line. The wave travels in the z direction. He defines the transverse Laplacian operator as the standard Laplacian though without the z partial derivatives. He then says that the transverse Laplacian for both E and H is zero for a TEM wave, i.e., where neither E nor H components in the z direction. This is equation (2-71) on p. 63.

I cannot see this for the life of me. Any thoughts would be appreciated.

BTW, I am not a student. I did EM graduate work about 25 years ago, but then my employer, who paid for that education, assigned me to other duties, and I stopped doing EM work. I am doing this now for fun.

Thanks.