let R be Ring; :: thesis: for V being RightMod of R
for v being Vector of V
for W being Submodule of V holds
( 0. V in v + W iff v in W )
let V be RightMod of R; :: thesis: for v being Vector of V
for W being Submodule of V holds
( 0. V in v + W iff v in W )
let v be Vector of V; :: thesis: for W being Submodule of V holds
( 0. V in v + W iff v in W )
let W be Submodule 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 0. V in v + W ; :: thesis: v in W
then consider u being Vector 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;

assume v in W ; :: thesis: 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