let V be ComplexLinearSpace; :: thesis: for V1 being Subset of V st the carrier of V = V1 holds
V1 is linearly-closed
let V1 be Subset of V; :: thesis: ( the carrier of V = V1 implies V1 is linearly-closed )
assume A1: the carrier of V = V1 ; :: thesis: V1 is linearly-closed
hence for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 ; :: according to CLVECT_1:def 4 :: thesis: for z being Complex
for v being VECTOR of V st v in V1 holds
z * v in V1
let z be Complex; :: thesis: for v being VECTOR of V st v in V1 holds
z * v in V1
let v be VECTOR of V; :: thesis: ( v in V1 implies z * v in V1 )
assume v in V1 ; :: thesis: z * v in V1
thus z * v in V1 by A1; :: thesis: verum