let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1 being Subspace of M
for W2, W3 being strict Subspace of M st W1 is Subspace of W2 holds
W1 + W3 is Subspace of W2 + W3
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for W1 being Subspace of M
for W2, W3 being strict Subspace of M st W1 is Subspace of W2 holds
W1 + W3 is Subspace of W2 + W3
let W1 be Subspace of M; :: thesis: for W2, W3 being strict Subspace of M st W1 is Subspace of W2 holds
W1 + W3 is Subspace of W2 + W3
let W2, W3 be strict Subspace of M; :: thesis: ( W1 is Subspace of W2 implies W1 + W3 is Subspace of W2 + W3 )
assume A1: W1 is Subspace of W2 ; :: thesis: W1 + W3 is Subspace of W2 + W3
(W1 + W3) + (W2 + W3) = (W1 + W3) + (W3 + W2) by Lm1
.= ((W1 + W3) + W3) + W2 by Th6
.= (W1 + (W3 + W3)) + W2 by Th6
.= (W1 + W3) + W2 by Lm3
.= W1 + (W3 + W2) by Th6
.= W1 + (W2 + W3) by Lm1
.= (W1 + W2) + W3 by Th6
.= W2 + W3 by A1, Th8 ;
hence W1 + W3 is Subspace of W2 + W3 by Th8; :: thesis: verum