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 )

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

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

( - u in W iff v + W = (v + (- u)) + W )
by Th53;
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;then - (- u) in W by Th22;

hence u in W by RLVECT_1:17; :: thesis: verum

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