  We prove a moving lemma for higher Chow groups with modulus, in the sense of
Binda-Kerz-Saito, of projective schemes when the modulus is given by a very
ample divisor. This provides one of the first cases of moving lemmas for cycles
with modulus, not covered by the additive higher Chow groups. We apply this to
prove a contravariant functoriality of higher Chow groups with modulus. We use
our moving techniques to show that the higher Chow groups of a line bundle over
a scheme, with the 0-section as the modulus, vanishes.
