reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
reserve M for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF;
reserve W,W1,W2,W3 for Subspace of M;
reserve u,u1,u2,v,v1,v2 for Element of M;
reserve X,Y for set, x,y,y1,y2 for object;

theorem Th20:
  (0).M /\ W = (0).M & W /\ (0).M = (0).M
proof
  0.M in W by VECTSP_4:17;
  then 0.M in the carrier of W by STRUCT_0:def 5;
  then {0.M} c= the carrier of W by ZFMISC_1:31;
  then
A1: {0.M} /\ (the carrier of W) = {0.M} by XBOOLE_1:28;
A2: the carrier of (0).M /\ W = (the carrier of (0).M) /\ (the carrier of W)
  by Def2
    .= {0.M} /\ (the carrier of W) by VECTSP_4:def 3;
  hence (0).M /\ W = (0).M by A1,VECTSP_4:def 3;
  thus thesis by A2,A1,VECTSP_4:def 3;
end;
