
theorem Th33:
  for K be add-associative right_zeroed right_complementable
  associative well-unital distributive non empty doubleLoopStr for V be
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital non empty
  ModuleStr over K for f be linear-Functional of V holds
  ker f is linearly-closed
proof
  let F be add-associative right_zeroed right_complementable associative
  well-unital distributive non empty doubleLoopStr;
  let V be add-associative right_zeroed right_complementable
  vector-distributive
  scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over F;
  let f be linear-Functional of V;
  set V1 = ker f;
  thus for v,u be Vector of V st v in V1 & u in V1 holds v + u in V1
  proof
    let v,u be Vector of V;
    assume v in V1 & u in V1;
    then
    (ex a be Vector of V st a = v & f.a= 0.F )& ex b be Vector of V st b =
    u & f.b= 0.F;
    then f.(v+u) = 0.F+0.F by VECTSP_1:def 20
      .= 0.F by RLVECT_1:4;
    hence thesis;
  end;
  let a be Element of F;
  let v be Vector of V;
  assume v in V1;
  then ex w be Vector of V st w=v & f.w=0.F;
  then f.(a*v) = a * 0.F by HAHNBAN1:def 8
    .=0.F;
  hence thesis;
end;
