
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_sup C c= lim_sup A /\ lim_sup
  B
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_sup C;
  for n being Nat ex k being Nat st x in B.(n+k)
  proof
    let n be Nat;
    consider k being Nat such that
A3: x in C.(n+k) by A2,Th5;
    take k;
    x in A.(n+k) /\ B.(n+k) by A1,A3;
    hence thesis by XBOOLE_0:def 4;
  end;
  then
A4: x in lim_sup B by Th5;
  for n being Nat ex k being Nat st x in A.(n+k)
  proof
    let n be Nat;
    consider k being Nat such that
A5: x in C.(n+k) by A2,Th5;
    take k;
    x in A.(n+k) /\ B.(n+k) by A1,A5;
    hence thesis by XBOOLE_0:def 4;
  end;
  then x in lim_sup A by Th5;
  hence thesis by A4,XBOOLE_0:def 4;
end;
