let V be ComplexLinearSpace; :: thesis: for v being VECTOR of V
for W being Subspace of V holds
( - v in v + W iff v in W )
let v be VECTOR of V; :: thesis: for W being Subspace of V holds
( - v in v + W iff v in W )
let W be Subspace of V; :: thesis: ( - v in v + W iff v in W )
A1: ( ( v in W implies (- 1r ) * v in v + W ) & (- 1r ) * v = - v ) by Th4, Th79;
( - 1r <> 1r & (- 1r ) * v in v + W implies v in W ) by Th78;
hence ( - v in v + W iff v in W ) by A1; :: thesis: verum