let R be Ring; :: thesis: for V being RightMod of R
for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1
let V be RightMod of R; :: thesis: for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1
let W1, W2 be Submodule of V; :: thesis: ( V is_the_direct_sum_of W1,W2 implies V is_the_direct_sum_of W2,W1 )
assume A1: V is_the_direct_sum_of W1,W2 ; :: thesis: V is_the_direct_sum_of W2,W1
then W1 /\ W2 = (0). V by Def4;
then A2: W2 /\ W1 = (0). V by Th18;
RightModStr(# the carrier of V,the U7 of V,the ZeroF of V,the rmult of V #) = W1 + W2 by A1, Def4;
then RightModStr(# the carrier of V,the U7 of V,the ZeroF of V,the rmult of V #) = W2 + W1 by Lm1;
hence V is_the_direct_sum_of W2,W1 by A2, Def4; :: thesis: verum