let V be RealLinearSpace; :: thesis: for u, v being VECTOR of V
for W being Subspace of V holds
( u + v in v + W iff u in W )
let u, v be VECTOR 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 + v in v + W ; :: thesis: u in W
then consider v1 being VECTOR of V such that
A1: u + v = v + v1 and
A2: v1 in W ;
thus u in W by A1, A2, RLVECT_1:21; :: thesis: verum

end;

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