reserve n for Nat;

theorem Th34: :: (2)
  for A, B being SetSequence of the carrier of TOP-REAL 2 st for i
  being Nat holds A.i c= B.i holds Lim_sup A c= Lim_sup B
proof
  let A, B be SetSequence of the carrier of TOP-REAL 2;
  assume
A1: for i being Nat holds A.i c= B.i;
  Lim_sup A c= Lim_sup B
  proof
    let x be object;
    assume x in Lim_sup A;
    then consider A1 being subsequence of A such that
A2: x in Lim_inf A1 by Def2;
    consider D being subsequence of B such that
A3: for i being Nat holds A1.i c= D.i by A1,Th32;
    Lim_inf A1 c= Lim_inf D by A3,Th17;
    hence thesis by A2,Def2;
  end;
  hence thesis;
end;
