let S be Subset of V; :: thesis: ( S = V1 /\ V2 implies S is linearly-closed )
assume A1: S = V1 /\ V2 ; :: thesis: S is linearly-closed
thus for v, u being VECTOR of V st v in S & u in S holds
v + u in S :: according to ZMODUL01:def 8 :: thesis: for a being Integer
for v being VECTOR of V st v in S holds
a * v in S

proof

let v, u be VECTOR of V; :: thesis: ( v in S & u in S implies v + u in S )
assume A2: ( v in S & u in S ) ; :: thesis: v + u in S
then ( v in V2 & u in V2 ) by A1, XBOOLE_0:def 4;
then A3: v + u in V2 by Def8;
( v in V1 & u in V1 ) by A2, A1, XBOOLE_0:def 4;
then v + u in V1 by Def8;
hence v + u in S by A3, A1, XBOOLE_0:def 4; :: thesis: verum

end;

let a be Integer; :: thesis: for v being VECTOR of V st v in S holds
a * v in S
let v be VECTOR of V; :: thesis: ( v in S implies a * v in S )
assume A4: v in S ; :: thesis: a * v in S
then v in V2 by A1, XBOOLE_0:def 4;
then A5: a * v in V2 by Def8;
v in V1 by A4, A1, XBOOLE_0:def 4;
then a * v in V1 by Def8;
hence a * v in S by A5, A1, XBOOLE_0:def 4; :: thesis: verum