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 strict VectSp-like Abelian add-associative right_zeroed VectSpStr of GF
for W being strict Subspace of V st ( for v being Element of V holds v in W ) holds
W = V
let V be non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for W being strict Subspace of V st ( for v being Element of V holds v in W ) holds
W = V
let W be strict Subspace of V; :: thesis: ( ( for v being Element of V holds v in W ) implies W = V )
assume for v being Element of V holds v in W ; :: thesis: W = V
then A1: for v being Element of V holds
( v in W iff v in V ) by RLVECT_1:3;
V is Subspace of V by Th32;
hence W = V by A1, Th38; :: thesis: verum