  This the first of a set of three papers about the Compression Theorem: if M^m
is embedded in Q^q X R with a normal vector field and if q-m > 0, then the
given vector field can be straightened (ie, made parallel to the given R
direction) by an isotopy of M and normal field in Q X R. The theorem can be
deduced from Gromov's theorem on directed embeddings [M Gromov, Partial
differential relations, Springer-Verlag (1986); 2.4.5 C'] and is implicit in
the preceeding discussion. Here we give a direct proof that leads to an
explicit description of the finishing embedding. In the second paper in the
series we give a proof in the spirit of Gromov's proof and in the third part we
give applications.
