reserve n for Nat;

theorem :: (3)
  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 A \/ Lim_sup B c=
  Lim_sup C
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;
A2: for i being Nat holds B.i c= C.i
  proof
    let i be Nat;
    C.i = A.i \/ B.i by A1;
    hence thesis by XBOOLE_1:7;
  end;
A3: for i being Nat holds A.i c= C.i
  proof
    let i be Nat;
    C.i = A.i \/ B.i by A1;
    hence thesis by XBOOLE_1:7;
  end;
  thus Lim_sup A \/ Lim_sup B c= Lim_sup C
  proof
    let x be object;
    assume
A4: x in Lim_sup A \/ Lim_sup B;
    per cases by A4,XBOOLE_0:def 3;
    suppose
      x in Lim_sup A;
      then consider A1 being subsequence of A such that
A5:   x in Lim_inf A1 by Def2;
      consider C1 being subsequence of C such that
A6:   for i being Nat holds A1.i c= C1.i by A3,Th32;
      Lim_inf A1 c= Lim_inf C1 by A6,Th17;
      hence thesis by A5,Def2;
    end;
    suppose
      x in Lim_sup B;
      then consider B1 being subsequence of B such that
A7:   x in Lim_inf B1 by Def2;
      consider C1 being subsequence of C such that
A8:   for i being Nat holds B1.i c= C1.i by A2,Th32;
      Lim_inf B1 c= Lim_inf C1 by A8,Th17;
      hence thesis by A7,Def2;
    end;
  end;
end;
