let R be Ring; :: thesis: for V being RightMod of R
for V1, V2 being Subset of V st V1 is linearly-closed & V2 is linearly-closed holds
V1 /\ V2 is linearly-closed
let V be RightMod of R; :: thesis: for V1, V2 being Subset of V st V1 is linearly-closed & V2 is linearly-closed holds
V1 /\ V2 is linearly-closed
let V1, V2 be Subset of V; :: thesis: ( V1 is linearly-closed & V2 is linearly-closed implies V1 /\ V2 is linearly-closed )
assume A1: ( V1 is linearly-closed & V2 is linearly-closed ) ; :: thesis: V1 /\ V2 is linearly-closed
thus for v, u being Vector of V st v in V1 /\ V2 & u in V1 /\ V2 holds
v + u in V1 /\ V2 :: according to RMOD_2:def 1 :: thesis: for a being Scalar of R
for v being Vector of V st v in V1 /\ V2 holds
v * a in V1 /\ V2

proof

let v, u be Vector of V; :: thesis: ( v in V1 /\ V2 & u in V1 /\ V2 implies v + u in V1 /\ V2 )
assume ( v in V1 /\ V2 & u in V1 /\ V2 ) ; :: thesis: v + u in V1 /\ V2
then ( v in V1 & v in V2 & u in V1 & u in V2 ) by XBOOLE_0:def 4;
then ( v + u in V1 & v + u in V2 ) by A1, Def1;
hence v + u in V1 /\ V2 by XBOOLE_0:def 4; :: thesis: verum

end;

let a be Scalar of R; :: thesis: for v being Vector of V st v in V1 /\ V2 holds
v * a in V1 /\ V2
let v be Vector of V; :: thesis: ( v in V1 /\ V2 implies v * a in V1 /\ V2 )
assume v in V1 /\ V2 ; :: thesis: v * a in V1 /\ V2
then ( v in V1 & v in V2 ) by XBOOLE_0:def 4;
then ( v * a in V1 & v * a in V2 ) by A1, Def1;
hence v * a in V1 /\ V2 by XBOOLE_0:def 4; :: thesis: verum