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 W being Subspace of V st x in W holds

x in V

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 W being Subspace of V st x in W holds

x in V

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 W being Subspace of V st x in W holds

x in V

let W be Subspace of V; :: thesis: ( x in W implies x in V )

assume x in W ; :: thesis: x in V

then A1: x in the carrier of W ;

the carrier of W c= the carrier of V by Def2;

hence x in V by A1; :: 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 W being Subspace of V st x in W holds

x in V

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 W being Subspace of V st x in W holds

x in V

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 W being Subspace of V st x in W holds

x in V

let W be Subspace of V; :: thesis: ( x in W implies x in V )

assume x in W ; :: thesis: x in V

then A1: x in the carrier of W ;

the carrier of W c= the carrier of V by Def2;

hence x in V by A1; :: thesis: verum