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 + v 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

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

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

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

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

thus ( u + v in v + W implies u in W ) :: thesis: ( u in W implies u + v in v + W )

hence u + v in v + W ; :: thesis: verum

for u, v being Element of V

for W being Subspace of V holds

( u + v 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

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

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

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

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

thus ( u + v in v + W implies u in W ) :: thesis: ( u in W implies u + v in v + W )

proof

assume
u in W
; :: thesis: u + v in v + W
assume
u + v in v + W
; :: thesis: u in W

then ex v1 being Element of V st

( u + v = v + v1 & v1 in W ) ;

hence u in W by RLVECT_1:8; :: thesis: verum

end;then ex v1 being Element of V st

( u + v = v + v1 & v1 in W ) ;

hence u in W by RLVECT_1:8; :: thesis: verum

hence u + v in v + W ; :: thesis: verum