reserve x,y,y1,y2 for object;

theorem
  for GF be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr, V be Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital non empty
  ModuleStr over GF, V1,V2 be Subset of V st V1 is linearly-closed & V2 is
  linearly-closed holds V1 /\ V2 is linearly-closed
proof
  let GF be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr, V be Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital non empty
  ModuleStr over GF, V1,V2 be Subset of V;
  assume that
A1: V1 is linearly-closed and
A2: V2 is linearly-closed;
  thus for v,u being Element of V st v in V1 /\ V2 & u in V1 /\ V2 holds v + u
  in V1 /\ V2
  proof
    let v,u be Element of V;
    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 be Element of GF;
  let v be Element of 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;
