theorem :: ZMODUL01:135

for R being Ring
for V being LeftMod of R
for W1, W2 being Submodule of V st V = W1 + W2 & ex v being Vector of V st
for v1, v2, u1, u2 being Vector of V st v1 + v2 = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 ) holds
V is_the_direct_sum_of W1,W2