reserve x for set,
  K for Ring,
  r for Scalar of K,
  V for LeftMod of K,
  a,b,a1,a2 for Vector of V,
  A,A1,A2 for Subset of V,
  l for Linear_Combination of A,
  W for Subspace of V,
  Li for FinSequence of Submodules(V);

theorem Th6:
  A is base & A = A1 \/ A2 & A1 misses A2 implies
  V is_the_direct_sum_of Lin A1,Lin A2
proof
  assume that
A1: A is base and
A2: A = A1 \/ A2 and
A3: A1 misses A2;
  set W=the ModuleStr of V;
A4: A is linearly-independent by A1,VECTSP_7:def 3;
  Lin A = W by A1,VECTSP_7:def 3;
  then
A5: W = Lin A1 + Lin A2 by A2,MOD_3:12;
  Lin A1 /\ Lin A2 = (0).V by A2,A3,A4,Th5;
  hence thesis by A5,VECTSP_5:def 4;
end;
