let V be RealLinearSpace; :: thesis: for V1 being Subset of V st V1 is linearly-closed holds
for v being VECTOR of V st v in V1 holds
- v in V1
let V1 be Subset of V; :: thesis: ( V1 is linearly-closed implies for v being VECTOR of V st v in V1 holds
- v in V1 )
assume A1: V1 is linearly-closed ; :: thesis: for v being VECTOR of V st v in V1 holds
- v in V1
let v be VECTOR of V; :: thesis: ( v in V1 implies - v in V1 )
assume v in V1 ; :: thesis: - v in V1
then (- jj) * v in V1 by A1;
hence - v in V1 by RLVECT_1:16; :: thesis: verum