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 Th21:
  for W being strict Subspace of M holds (Omega).M /\ W = W & W /\
  (Omega).M = W
proof
  let W be strict Subspace of M;
A1: the carrier of (Omega).M /\ W = (the carrier of the ModuleStr of M) /\ (
the carrier of W) & the carrier of W c= the carrier of M by Def2,VECTSP_4:def 2
  ;
  hence (Omega).M /\ W = W by VECTSP_4:29,XBOOLE_1:28;
  thus thesis by A1,VECTSP_4:29,XBOOLE_1:28;
end;
