reserve n for Nat;

theorem :: (4)
  for A, B, C being SetSequence of the carrier of TOP-REAL 2 st for i
  being Nat holds C.i = A.i /\ B.i holds Lim_sup C c= Lim_sup A /\
  Lim_sup B
proof
  let A, B, C be SetSequence of the carrier of TOP-REAL 2;
  assume
A1: for i being Nat holds C.i = A.i /\ B.i;
  let x be object;
  assume x in Lim_sup C;
  then consider C1 being subsequence of C such that
A2: x in Lim_inf C1 by Def2;
  for i being Nat holds C.i c= B.i
  proof
    let i be Nat;
    C.i = A.i /\ B.i by A1;
    hence thesis by XBOOLE_1:17;
  end;
  then consider E1 being subsequence of B such that
A3: for i being Nat holds C1.i c= E1.i by Th32;
  Lim_inf C1 c= Lim_inf E1 by A3,Th17;
  then
A4: x in Lim_sup B by A2,Def2;
  for i being Nat holds C.i c= A.i
  proof
    let i be Nat;
    C.i = A.i /\ B.i by A1;
    hence thesis by XBOOLE_1:17;
  end;
  then consider D1 being subsequence of A such that
A5: for i being Nat holds C1.i c= D1.i by Th32;
  Lim_inf C1 c= Lim_inf D1 by A5,Th17;
  then x in Lim_sup A by A2,Def2;
  hence thesis by A4,XBOOLE_0:def 4;
end;
