V.19 Proposition V.19
If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole.
Proof
If a:c=a′:c′ with a′<a, c′<c, apply V.17 (separation)
to obtain (a−a′):a′=(c−c′):c′, and V.11 / V.16 to
re-express as (a−a′):(c−c′)=a:c.
lines 74–74 in main.tex