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;

theorem Th39:
  for L being Linear_Compl of W holds W + L = the ModuleStr of V &
  L + W = the ModuleStr of V
proof
  let L be Linear_Compl of W;
  V is_the_direct_sum_of W,L by Th38;
  hence W + L = the ModuleStr of V;
  hence thesis by Lm1;
end;
