
theorem
  for X being set, A, B, C being SetSequence of X
   st for n being Nat holds C.n = A.n \/ B.n
   holds lim_inf A \/ lim_inf B c= lim_inf C
proof
  let X be set, A, B, C be SetSequence of X;
  assume
A1: for n being Nat holds C.n = A.n \/ B.n;
  let x be object;
  assume
A2: x in lim_inf A \/ lim_inf B;
  per cases by A2,XBOOLE_0:def 3;
  suppose
    x in lim_inf A;
    then consider n being Nat such that
A3: for k being Nat holds x in A.(n+k) by Th4;
    for k being Nat holds x in C.(n+k)
    proof
      let k be Nat;
      x in A.(n+k) by A3;
      then x in A.(n+k) \/ B.(n+k) by XBOOLE_0:def 3;
      hence thesis by A1;
    end;
    hence thesis by Th4;
  end;
  suppose
    x in lim_inf B;
    then consider n being Nat such that
A4: for k being Nat holds x in B.(n+k) by Th4;
    for k being Nat holds x in C.(n+k)
    proof
      let k be Nat;
      x in B.(n+k) by A4;
      then x in A.(n+k) \/ B.(n+k) by XBOOLE_0:def 3;
      hence thesis by A1;
    end;
    hence thesis by Th4;
  end;
end;
