reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;

theorem
  V1 is linearly-closed & V2 is linearly-closed implies V1 /\ V2 is
  linearly-closed
proof
  assume that
A1: V1 is linearly-closed and
A2: V2 is linearly-closed;
  thus for v,u st v in V1 /\ V2 & u in V1 /\ V2 holds v + u in V1 /\ V2
  proof
    let v,u;
    assume
A3: v in V1 /\ V2 & 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 thesis by A4,XBOOLE_0:def 4;
  end;
  let a,v;
  assume
A5: v in V1 /\ V2;
  then v in V2 by XBOOLE_0:def 4;
  then
A6: a * v in V2 by A2;
  v in V1 by A5,XBOOLE_0:def 4;
  then a * v in V1 by A1;
  hence thesis by A6,XBOOLE_0:def 4;
end;
