let V be Z_Module; :: thesis: ( V is_the_direct_sum_of (0). V, (Omega). V & V is_the_direct_sum_of (Omega). V, (0). V )
A1: ( ((0). V) + ((Omega). V) = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) & (0). V = ((0). V) /\ ((Omega). V) ) by Th99, Th107;
hence V is_the_direct_sum_of (0). V, (Omega). V by Def17; :: thesis: V is_the_direct_sum_of (Omega). V, (0). V
thus V is_the_direct_sum_of (Omega). V, (0). V by A1, Def17; :: thesis: verum