reserve V for RealLinearSpace;
reserve W,W1,W2,W3 for Subspace of V;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a,a1,a2 for Real;
reserve X,Y,x,y,y1,y2 for set;

theorem Th18:
  (0).V /\ W = (0).V & W /\ (0).V = (0).V
proof
  0.V in W by RLSUB_1:17;
  then 0.V in the carrier of W by STRUCT_0:def 5;
  then {0.V} c= the carrier of W by ZFMISC_1:31;
  then
A1: {0.V} /\ (the carrier of W) = {0.V} by XBOOLE_1:28;
  the carrier of (0).V /\ W = (the carrier of (0).V) /\ (the carrier of W)
  by Def2
    .= {0.V} /\ (the carrier of W) by RLSUB_1:def 3;
  hence (0).V /\ W = (0).V by A1,RLSUB_1:def 3;
  hence thesis by Th14;
end;
