let x be object ; :: thesis: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr

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 st x in W1 & W1 is Subspace of W2 holds

x in W2

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 st x in W1 & W1 is Subspace of W2 holds

x in 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 st x in W1 & W1 is Subspace of W2 holds

x in W2

let W1, W2 be Subspace of V; :: thesis: ( x in W1 & W1 is Subspace of W2 implies x in W2 )

assume ( x in W1 & W1 is Subspace of W2 ) ; :: thesis: x in W2

then ( x in the carrier of W1 & the carrier of W1 c= the carrier of W2 ) by Def2;

hence x in W2 ; :: thesis: verum

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 st x in W1 & W1 is Subspace of W2 holds

x in W2

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 st x in W1 & W1 is Subspace of W2 holds

x in 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 st x in W1 & W1 is Subspace of W2 holds

x in W2

let W1, W2 be Subspace of V; :: thesis: ( x in W1 & W1 is Subspace of W2 implies x in W2 )

assume ( x in W1 & W1 is Subspace of W2 ) ; :: thesis: x in W2

then ( x in the carrier of W1 & the carrier of W1 c= the carrier of W2 ) by Def2;

hence x in W2 ; :: thesis: verum