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, W2, W3 being Subspace of M st W1 is Subspace of W3 & W2 is Subspace of W3 holds
W1 + W2 is Subspace of 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, W2, W3 being Subspace of M st W1 is Subspace of W3 & W2 is Subspace of W3 holds
W1 + W2 is Subspace of W3

let W1, W2, W3 be Subspace of M; :: thesis: ( W1 is Subspace of W3 & W2 is Subspace of W3 implies W1 + W2 is Subspace of W3 )
assume A1: ( W1 is Subspace of W3 & W2 is Subspace of W3 ) ; :: thesis: W1 + W2 is Subspace of W3
now :: thesis: for v being Element of M st v in W1 + W2 holds
v in W3
let v be Element of M; :: thesis: ( v in W1 + W2 implies v in W3 )
assume v in W1 + W2 ; :: thesis: v in W3
then consider v1, v2 being Element of M such that
A2: ( v1 in W1 & v2 in W2 ) and
A3: v = v1 + v2 by Th1;
( v1 in W3 & v2 in W3 ) by ;
hence v in W3 by ; :: thesis: verum
end;
hence W1 + W2 is Subspace of W3 by VECTSP_4:28; :: thesis: verum