reserve R for Ring,
  V for RightMod of R,
  W,W1,W2,W3 for Submodule of V,
  u,u1, u2,v,v1,v2 for Vector of V,
  x,y,y1,y2 for object;
reserve C1 for Coset of W1;
reserve C2 for Coset of W2;

theorem Th43:
  for V being RightMod of R, W1,W2 being Submodule of V, v,v1,v2,
u1,u2 being Vector of V holds V is_the_direct_sum_of W1,W2 & v = v1 + v2 & v =
  u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 implies v1 = u1 & v2 = u2
proof
  let V be RightMod of R, W1,W2 be Submodule of V, v,v1,v2,u1,u2 be Vector of
  V;
  reconsider C2 = v1 + W2 as Coset of W2 by RMOD_2:def 6;
  reconsider C1 = the carrier of W1 as Coset of W1 by RMOD_2:70;
A1: v1 in C2 by RMOD_2:44;
  assume V is_the_direct_sum_of W1,W2;
  then consider u being Vector of V such that
A2: C1 /\ C2 = {u} by Th41;
  assume that
A3: v = v1 + v2 & v = u1 + u2 and
A4: v1 in W1 and
A5: u1 in W1 and
A6: v2 in W2 & u2 in W2;
A7: v2 - u2 in W2 by A6,RMOD_2:23;
  v1 in C1 by A4;
  then v1 in C1 /\ C2 by A1,XBOOLE_0:def 4;
  then
A8: v1 = u by A2,TARSKI:def 1;
  u1 = (v1 + v2) - u2 by A3,Lm15
    .= v1 + (v2 - u2) by RLVECT_1:def 3;
  then
A9: u1 in C2 by A7;
  u1 in C1 by A5;
  then
A10: u1 in C1 /\ C2 by A9,XBOOLE_0:def 4;
  hence v1 = u1 by A2,A8,TARSKI:def 1;
  u1 = u by A10,A2,TARSKI:def 1;
  hence thesis by A3,A8,RLVECT_1:8;
end;
