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

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

hence
W1 + W2 is Subspace of W3
by VECTSP_4:28; :: thesis: verumv 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 A1, A2, VECTSP_4:8;

hence v in W3 by A3, VECTSP_4:20; :: thesis: verum

end;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 A1, A2, VECTSP_4:8;

hence v in W3 by A3, VECTSP_4:20; :: thesis: verum