let V be ComplexLinearSpace; :: thesis: for v being VECTOR of V
for W being Subspace of V holds
( 0. V in v + W iff v in W )
let v be VECTOR 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 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, RLVECT_1:def 10;
hence v in W by A2, Th41; :: thesis: verum

end;

assume v in W ; :: thesis: 0. V in v + W
then A3: - v in W by Th41;
0. V = v - v by RLVECT_1:15
.= v + (- v) ;
hence 0. V in v + W by A3; :: thesis: verum