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 that
A1: V1 is linearly-closed and
A2: 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 A3: ( v in V1 /\ V2 & u in V1 /\ V2 ) ; :: thesis: v + u in V1 /\ V2
then ( v in V2 & u in V2 ) by XBOOLE_0:def 4;
then A4: v + u in V2 by A2;
( v in V1 & u in V1 ) by A3, XBOOLE_0:def 4;
then v + u in V1 by A1;
hence v + u in V1 /\ V2 by A4, 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 A5: v in V1 /\ V2 ; :: thesis: v * a in V1 /\ V2
then v in V2 by XBOOLE_0:def 4;
then A6: v * a in V2 by A2;
v in V1 by A5, XBOOLE_0:def 4;
then v * a in V1 by A1;
hence v * a in V1 /\ V2 by A6, XBOOLE_0:def 4; :: thesis: verum