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 VectSp-like Abelian add-associative right_zeroed VectSpStr of GF
for v being Element of V
for W being Subspace of V holds
( 0. V in v + W iff v + W = the carrier of W )
let V be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for v being Element of V
for W being Subspace of V holds
( 0. V in v + W iff v + W = the carrier of W )
let v be Element of V; :: thesis: for W being Subspace of V holds
( 0. V in v + W iff v + W = the carrier of W )
let W be Subspace of V; :: thesis: ( 0. V in v + W iff v + W = the carrier of W )
( ( 0. V in v + W implies v in W ) & ( v in W implies 0. V in v + W ) & ( v in W implies v + W = the carrier of W ) & ( v + W = the carrier of W implies v in W ) ) by Lm4, Th58;
hence ( 0. V in v + W iff v + W = the carrier of W ) ; :: thesis: verum