let GF be Field; :: thesis: for V being VectSp of GF
for v being Element of V
for W being Subspace of V holds
( v in W iff (- v) + W = the carrier of W )
let V be VectSp of GF; :: thesis: for v being Element of V
for W being Subspace of V holds
( v in W iff (- v) + W = the carrier of W )
let v be Element of V; :: thesis: for W being Subspace of V holds
( v in W iff (- v) + W = the carrier of W )
let W be Subspace of V; :: thesis: ( v in W iff (- v) + W = the carrier of W )
- (1_ GF) <> 0. GF by VECTSP_2:3;
then ( v in W iff ((- (1_ GF)) * v) + W = the carrier of W ) by Th50, Th51;
hence ( v in W iff (- v) + W = the carrier of W ) by VECTSP_1:14; :: thesis: verum