let F be RFunctional of V; :: thesis: ( F is Real_homogeneous implies F is 0-preserving )
assume A1: F is Real_homogeneous ; :: thesis: F is 0-preserving
A2: 0. F_Complex = [**0,0**] by COMPLFLD:7;
thus F . (0. V) = F . ((0. F_Complex) * (0. V)) by VECTSP_1:14
.= 0 * (F . (0. V)) by A1, A2
.= 0 ; :: according to HAHNBAN1:def 15 :: thesis: verum