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

( v - u in v + W iff u in 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

( v - u in v + W iff u in W )

let u, v be Element of V; :: thesis: for W being Subspace of V holds

( v - u in v + W iff u in W )

let W be Subspace of V; :: thesis: ( v - u in v + W iff u in W )

A1: v - u = (- u) + v ;

A2: ( - u in W implies - (- u) in W ) by Th22;

( u in W implies - u in W ) by Th22;

hence ( v - u in v + W iff u in W ) by A1, A2, Th60, RLVECT_1:17; :: thesis: verum

for u, v being Element of V

for W being Subspace of V holds

( v - u in v + W iff u in 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

( v - u in v + W iff u in W )

let u, v be Element of V; :: thesis: for W being Subspace of V holds

( v - u in v + W iff u in W )

let W be Subspace of V; :: thesis: ( v - u in v + W iff u in W )

A1: v - u = (- u) + v ;

A2: ( - u in W implies - (- u) in W ) by Th22;

( u in W implies - u in W ) by Th22;

hence ( v - u in v + W iff u in W ) by A1, A2, Th60, RLVECT_1:17; :: thesis: verum