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

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