let V be RealLinearSpace; :: 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 RLSUB_1:def 1 :: thesis: for a being Real
for v being VECTOR of V st v in V1 holds
a * v in V1
let a be Real; :: thesis: for v being VECTOR of V st v in V1 holds
a * v in V1
let v be VECTOR of V; :: thesis: ( v in V1 implies a * v in V1 )
assume v in V1 ; :: thesis: a * v in V1
thus a * v in V1 by A1; :: thesis: verum