let R be Ring; :: thesis: for V being RightMod of R
for V1 being Subset of V st the carrier of V = V1 holds
V1 is linearly-closed
let V be RightMod of R; :: 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 RMOD_2:def 1 :: thesis: for a being Scalar of R
for v being Vector of V st v in V1 holds
v * a in V1
let a be Scalar of R; :: thesis: for v being Vector of V st v in V1 holds
v * a in V1
let v be Vector of V; :: thesis: ( v in V1 implies v * a in V1 )
assume v in V1 ; :: thesis: v * a in V1
thus v * a in V1 by A1; :: thesis: verum