let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for W1, W2 being Subspace of V holds 0. W1 = 0. W2

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for W1, W2 being Subspace of V holds 0. W1 = 0. W2

let W1, W2 be Subspace of V; :: thesis: 0. W1 = 0. W2

thus 0. W1 = 0. V by Def2

.= 0. W2 by Def2 ; :: thesis: verum

for W1, W2 being Subspace of V holds 0. W1 = 0. W2

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for W1, W2 being Subspace of V holds 0. W1 = 0. W2

let W1, W2 be Subspace of V; :: thesis: 0. W1 = 0. W2

thus 0. W1 = 0. V by Def2

.= 0. W2 by Def2 ; :: thesis: verum