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 v being Element of V

for W being Subspace of V holds

( 0. V in v + W iff v 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 v being Element of V

for W being Subspace of V holds

( 0. V in v + W iff v in W )

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

( 0. V in v + W iff v in W )

let W be Subspace of V; :: thesis: ( 0. V in v + W iff v in W )

thus ( 0. V in v + W implies v in W ) :: thesis: ( v in W implies 0. V in v + W )

then A3: - v in W by Th22;

0. V = v + (- v) by VECTSP_1:19;

hence 0. V in v + W by A3; :: thesis: verum

for v being Element of V

for W being Subspace of V holds

( 0. V in v + W iff v 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 v being Element of V

for W being Subspace of V holds

( 0. V in v + W iff v in W )

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

( 0. V in v + W iff v in W )

let W be Subspace of V; :: thesis: ( 0. V in v + W iff v in W )

thus ( 0. V in v + W implies v in W ) :: thesis: ( v in W implies 0. V in v + W )

proof

assume
v in W
; :: thesis: 0. V in v + W
assume
0. V in v + W
; :: thesis: v in W

then consider u being Element of V such that

A1: 0. V = v + u and

A2: u in W ;

v = - u by A1, VECTSP_1:16;

hence v in W by A2, Th22; :: thesis: verum

end;then consider u being Element of V such that

A1: 0. V = v + u and

A2: u in W ;

v = - u by A1, VECTSP_1:16;

hence v in W by A2, Th22; :: thesis: verum

then A3: - v in W by Th22;

0. V = v + (- v) by VECTSP_1:19;

hence 0. V in v + W by A3; :: thesis: verum