reserve n for Nat;

theorem
  for A, B being SetSequence of the carrier of TOP-REAL 2, C being
  SetSequence of the carrier of [: TOP-REAL 2, 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 be SetSequence of the carrier of TOP-REAL 2, C be SetSequence of
  the carrier of [: TOP-REAL 2, 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;
  x in the carrier of [: TOP-REAL 2, TOP-REAL 2 :] by A2;
  then x in [: the carrier of TOP-REAL 2, the carrier of TOP-REAL 2 :] by
BORSUK_1:def 2;
  then consider x1, x2 being object such that
A3: x = [x1, x2] by RELAT_1:def 1;
  consider A1, B1 being SetSequence of the carrier of TOP-REAL 2 such that
A4: A1 is subsequence of A and
A5: B1 is subsequence of B and
A6: for i being Nat holds C1.i = [: A1.i, B1.i :] by A1,Th37;
A7: x in [: Lim_inf A1, Lim_inf B1 :] by A2,A6,Th27;
  then x2 in Lim_inf B1 by A3,ZFMISC_1:87;
  then
A8: x2 in Lim_sup B by A5,Def2;
  x1 in Lim_inf A1 by A3,A7,ZFMISC_1:87;
  then x1 in Lim_sup A by A4,Def2;
  hence thesis by A3,A8,ZFMISC_1:87;
end;
