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;
reserve F for Field;
reserve V for VectSp of F;
reserve W for Subspace of V;
reserve W,W1,W2 for Subspace of V;
reserve W1,W2 for Subspace of M;
reserve W for Subspace of V;
reserve W1,W2 for Subspace of M;
reserve u,u1,u2,v for Element of M;
reserve C1 for Coset of W1;
reserve C2 for Coset of W2;

theorem
  for M being strict Abelian add-associative right_zeroed
  right_complementable vector-distributive scalar-distributive
  scalar-associative scalar-unital
   non empty ModuleStr over GF, W1,W2 being
  Subspace of M holds W1 + W2 = M iff for v being Element of M ex v1,v2 being
  Element of M st v1 in W1 & v2 in W2 & v = v1 + v2 by Lm16;
