reserve n for Nat;

theorem Th17:
  for T being non empty TopSpace, A, B being SetSequence of the
carrier of T st for i being Nat holds A.i c= B.i holds Lim_inf A c=
  Lim_inf B
proof
  let T be non empty TopSpace, A, B be SetSequence of the carrier of T;
  assume
A1: for i being Nat holds A.i c= B.i;
  let x be object;
  assume
A2: x in Lim_inf A;
  then reconsider p = x as Point of T;
  for G being a_neighborhood of p ex k being Nat st for m being
  Nat st m > k holds B.m meets G
  proof
    let G be a_neighborhood of p;
    consider k being Nat such that
A3: for m being Nat st m > k holds A.m meets G by A2,Def1;
    take k;
    let m1 be Nat;
    assume m1 > k;
    then A.m1 meets G by A3;
    hence thesis by A1,XBOOLE_1:63;
  end;
  hence thesis by Def1;
end;
