let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF holds
( M is_the_direct_sum_of (0). M, (Omega). M & M is_the_direct_sum_of (Omega). M, (0). M )
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: ( M is_the_direct_sum_of (0). M, (Omega). M & M is_the_direct_sum_of (Omega). M, (0). M )
( ((0). M) + ((Omega). M) = VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) & (0). M = ((0). M) /\ ((Omega). M) ) by Th13, Th25;
hence M is_the_direct_sum_of (0). M, (Omega). M by Def4; :: thesis: M is_the_direct_sum_of (Omega). M, (0). M
hence M is_the_direct_sum_of (Omega). M, (0). M by Lm17; :: thesis: verum