reserve n for Nat;

theorem
  for T being non empty TopSpace, A, B, C being SetSequence of the
carrier of T st for i being Nat holds C.i = A.i /\ B.i holds Lim_inf
  C c= Lim_inf A /\ Lim_inf B
proof
  let T be non empty TopSpace, A, B, C be SetSequence of the carrier of T;
  assume
A1: for i being Nat holds C.i = A.i /\ B.i;
  let x be object;
  assume
A2: x in Lim_inf C;
  then reconsider p = x as Point of T;
  for H being a_neighborhood of p ex k being Nat st for m being
  Nat st m > k holds B.m meets H
  proof
    let H be a_neighborhood of p;
    consider k being Nat such that
A3: for m being Nat st m > k holds C.m meets H by A2,Def1;
    take k;
    let m be Nat;
    assume m > k;
    then
A4: C.m meets H by A3;
    C.m = A.m /\ B.m by A1;
    hence thesis by A4,XBOOLE_1:17,63;
  end;
  then
A5: x in Lim_inf B by Def1;
  for H being a_neighborhood of p ex k being Nat st for m being
  Nat st m > k holds A.m meets H
  proof
    let H be a_neighborhood of p;
    consider k being Nat such that
A6: for m being Nat st m > k holds C.m meets H by A2,Def1;
    take k;
    let m be Nat;
    assume m > k;
    then
A7: C.m meets H by A6;
    C.m = A.m /\ B.m by A1;
    hence thesis by A7,XBOOLE_1:17,63;
  end;
  then x in Lim_inf A by Def1;
  hence thesis by A5,XBOOLE_0:def 4;
end;
