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 Th7:
  W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2
proof
  the carrier of W1 c= the carrier of W1 + W2 by Lm2;
  hence W1 is Subspace of W1 + W2 by VECTSP_4:27;
  the carrier of W2 c= the carrier of W2 + W1 by Lm2;
  then the carrier of W2 c= the carrier of W1 + W2 by Lm1;
  hence thesis by VECTSP_4:27;
end;
