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 u, v being Element of V
for W being Subspace of V holds
( u in W iff v + W = (v - u) + W )
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 u, v being Element of V
for W being Subspace of V holds
( u in W iff v + W = (v - u) + W )
let u, v be Element of V; :: thesis: for W being Subspace of V holds
( u in W iff v + W = (v - u) + W )
let W be Subspace of V; :: thesis: ( u in W iff v + W = (v - u) + W )
A1: ( - u in W implies u in W )

proof

assume - u in W ; :: thesis: u in W
then - (- u) in W by Th22;
hence u in W by RLVECT_1:17; :: thesis: verum

end;

( - u in W iff v + W = (v + (- u)) + W ) by Th53;
hence ( u in W iff v + W = (v - u) + W ) by A1, Th22; :: thesis: verum