reserve n for Nat;

theorem Th27:
  for A, B being SetSequence of the carrier of TOP-REAL n, C being
  SetSequence of the carrier of [: TOP-REAL n, TOP-REAL n :] st for i being
  Nat holds C.i = [:A.i, B.i:] holds [: Lim_inf A, Lim_inf B :] =
  Lim_inf C
proof
  let A, B be SetSequence of the carrier of TOP-REAL n, C be SetSequence of
  the carrier of [: TOP-REAL n, TOP-REAL n :];
  assume
A1: for i being Nat holds C.i = [:A.i, B.i:];
  thus [: Lim_inf A, Lim_inf B :] c= Lim_inf C
  proof
    let x be object;
    assume
A2: x in [: Lim_inf A, Lim_inf B :];
    then consider x1, x2 being object such that
A3: x1 in Lim_inf A and
A4: x2 in Lim_inf B and
A5: x = [x1,x2] by ZFMISC_1:def 2;
    reconsider p = x as Point of [: TOP-REAL n, TOP-REAL n :] by A2;
    reconsider x1, x2 as Point of TOP-REAL n by A3,A4;
    for G being a_neighborhood of p ex k being Nat st for m
    being Nat st m > k holds C.m meets G
    proof
      let G be a_neighborhood of p;
      G is a_neighborhood of [:{x1},{x2}:] by A5,Th11;
      then consider
      V being a_neighborhood of {x1}, W being a_neighborhood of x2
      such that
A6:   [:V,W:] c= G by BORSUK_1:25;
      consider k2 being Nat such that
A7:   for m being Nat st m > k2 holds B.m meets W by A4,Def1;
      V is a_neighborhood of x1 by CONNSP_2:8;
      then consider k1 being Nat such that
A8:   for m being Nat st m > k1 holds A.m meets V by A3,Def1;
      reconsider k = max (k1, k2) as Nat by TARSKI:1;
      take k;
      let m be Nat;
      assume
A9:   m > k;
      k >= k2 by XXREAL_0:25;
      then m > k2 by A9,XXREAL_0:2;
      then
A10:  B.m meets W by A7;
      k >= k1 by XXREAL_0:25;
      then m > k1 by A9,XXREAL_0:2;
      then A.m meets V by A8;
      then [: A.m, B.m :] meets [: V, W :] by A10,KURATO_0:2;
      then C.m meets [: V, W :] by A1;
      hence thesis by A6,XBOOLE_1:63;
    end;
    hence thesis by Def1;
  end;
  thus Lim_inf C c= [: Lim_inf A, Lim_inf B :]
  proof
    let x be object;
    assume
A11: x in Lim_inf C;
    then x in the carrier of [: TOP-REAL n, TOP-REAL n :];
    then
A12: x in [: the carrier of TOP-REAL n, the carrier of TOP-REAL n :] by
BORSUK_1:def 2;
    then reconsider p1 = x`1, p2 = x`2 as Point of TOP-REAL n by MCART_1:10;
    set H = the a_neighborhood of p2;
    set F = the a_neighborhood of p1;
A13: x = [p1,p2] by A12,MCART_1:21;
    for G being a_neighborhood of p2 ex k being Nat st for m
    being Nat st m > k holds B.m meets G
    proof
      let G be a_neighborhood of p2;
      consider k being Nat such that
A14:  for m being Nat st m > k holds C.m meets [: F, G :]
      by A11,A13,Def1;
      take k;
      let m be Nat;
      assume m > k;
      then C.m meets [: F, G :] by A14;
      then consider y being object such that
A15:  y in C.m and
A16:  y in [: F, G :] by XBOOLE_0:3;
      y in [:A.m, B.m:] by A1,A15;
      then
A17:  y`2 in B.m by MCART_1:10;
      y`2 in G by A16,MCART_1:10;
      hence thesis by A17,XBOOLE_0:3;
    end;
    then
A18: p2 in Lim_inf B by Def1;
    for G being a_neighborhood of p1 ex k being Nat st for m
    being Nat st m > k holds A.m meets G
    proof
      let G be a_neighborhood of p1;
      consider k being Nat such that
A19:  for m being Nat st m > k holds C.m meets [: G, H :]
      by A11,A13,Def1;
      take k;
      let m be Nat;
      assume m > k;
      then C.m meets [: G, H :] by A19;
      then consider y being object such that
A20:  y in C.m and
A21:  y in [: G, H :] by XBOOLE_0:3;
      y in [:A.m, B.m:] by A1,A20;
      then
A22:  y`1 in A.m by MCART_1:10;
      y`1 in G by A21,MCART_1:10;
      hence thesis by A22,XBOOLE_0:3;
    end;
    then p1 in Lim_inf A by Def1;
    hence thesis by A13,A18,ZFMISC_1:87;
  end;
end;
