let V be Z_Module; :: 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 A2: W2 /\ W1 = (0). V by Def17;
Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) = W1 + W2 by A1, Def17;
hence V is_the_direct_sum_of W2,W1 by A2, Def17; :: thesis: verum