let f be Functional of V; :: thesis: ( f is semi-homogeneous implies f is 0-preserving )
assume A4: f is semi-homogeneous ; :: thesis: f is 0-preserving
thus f . (0. V) = f . (0 * (0. V)) by RLVECT_1:10
.= 0 * (f . (0. V)) by A4, Def7
.= 0 ; :: according to HAHNBAN:def 7 :: thesis: verum