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
  A \/ Lim_inf B c= Lim_inf C
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 A \/ Lim_inf B;
  then reconsider p = x as Point of T;
  per cases by A2,XBOOLE_0:def 3;
  suppose
A3: x in Lim_inf A;
    for H being a_neighborhood of p ex k being Nat st for m
    being Nat st m > k holds C.m meets H
    proof
      let H be a_neighborhood of p;
      consider k being Nat such that
A4:   for m being Nat st m > k holds A.m meets H by A3,Def1;
      take k;
      let m be Nat;
      assume m > k;
      then
A5:   A.m meets H by A4;
      C.m = A.m \/ B.m by A1;
      hence thesis by A5,XBOOLE_1:7,63;
    end;
    hence thesis by Def1;
  end;
  suppose
A6: x in Lim_inf B;
    for H being a_neighborhood of p ex k being Nat st for m
    being Nat st m > k holds C.m meets H
    proof
      let H be a_neighborhood of p;
      consider k being Nat such that
A7:   for m being Nat st m > k holds B.m meets H by A6,Def1;
      take k;
      let m be Nat;
      assume m > k;
      then
A8:   B.m meets H by A7;
      C.m = A.m \/ B.m by A1;
      hence thesis by A8,XBOOLE_1:7,63;
    end;
    hence thesis by Def1;
  end;
end;
