let V be ComplexLinearSpace; :: thesis: for v being VECTOR of V
for W being Subspace of V holds
( v in W iff (- v) + W = the carrier of W )
let v be VECTOR 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 )
( ( v in W implies ((- 1r ) * v) + W = the carrier of W ) & ( ((- 1r ) * v) + W = the carrier of W implies v in W ) & (- 1r ) * v = - v ) by Th4, Th69, Th70;
hence ( v in W iff (- v) + W = the carrier of W ) ; :: thesis: verum