Questions about Deriving the Vanna Formula

In the below image, on step 5, they have the formula:

e^(-qt)*sqrt(T-t)* N'(d1)(d2/σ)


Step 6, they adjust the bolded part to:

e^(-qt)*sqrt(T-t)* N'(d1)(1-d1)

and I can't seem to figure out how they did it. I'd be grateful if anyone had a little bit of knowledge they'd want to share. (I included the part about vega formula below this image and a link to the actual PDF as well.)

upload_2022-1-7_8-29-48.png

upload_2022-1-7_8-35-57.png
upload_2022-1-7_8-29-48.png
upload_2022-1-7_8-35-57.png


https://acfr.aut.ac.nz/__data/assets/pdf_file/0017/185300/172496-K-Huang-Vanna-Volga_Auckland.pdf

Cheers,
NN
 
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