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_2:34;
then ( ( v in W implies ((- (1_ GF)) * v) + W = the carrier of W ) & ( ((- (1_ GF)) * v) + W = the carrier of W implies v in W ) & (- (1_ GF)) * v = - v ) by Th65, Th66, VECTSP_1:59;
hence ( v in W iff (- v) + W = the carrier of W ) ; :: thesis: verum