let x be set ; :: thesis: for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W being Subspace of V st x in W holds
x in V
let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of 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 VectSpStr of 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 by STRUCT_0:def 5;
the carrier of W c= the carrier of V by Def2;
hence x in V by A1, STRUCT_0:def 5; :: thesis: verum