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
for W1, W2 being Subspace of M st M is_the_direct_sum_of W1,W2 holds
M is_the_direct_sum_of W2,W1
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for W1, W2 being Subspace of M st M is_the_direct_sum_of W1,W2 holds
M is_the_direct_sum_of W2,W1
let W1, W2 be Subspace of M; :: thesis: ( M is_the_direct_sum_of W1,W2 implies M is_the_direct_sum_of W2,W1 )
assume A1: M is_the_direct_sum_of W1,W2 ; :: thesis: M is_the_direct_sum_of W2,W1
then VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) = W1 + W2 by Def4;
then A2: VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) = W2 + W1 by Lm1;
W2 /\ W1 = (0). M by A1, Def4;
hence M is_the_direct_sum_of W2,W1 by A2, Def4; :: thesis: verum