reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th25:
  for t being Point of Y, A being Subset of X st A is compact for
  G being a_neighborhood of [:A,{t}:] ex V being a_neighborhood of A, W being
  a_neighborhood of t st [:V,W:] c= G
proof
  let t be Point of Y, A be Subset of X such that
A1: A is compact;
  let G be a_neighborhood of [:A,{t}:];
A2: the carrier of [:X,Y:] = [:the carrier of X, the carrier of Y:] by Def2;
  now
    per cases;
    suppose
A3:   A <> {} X;
      [:A,{t}:] c= Int G by CONNSP_2:def 2;
      then [:A,{t}:] c= union Base-Appr G by Th14;
      then Base-Appr G is Cover of [:A,{t}:] by SETFAM_1:def 11;
      then consider K being Subset-Family of [:X,Y:] such that
A4:   K c= Base-Appr G and
A5:   K is Cover of [:A,{t}:] and
A6:   for c being set st c in K holds c meets [:A,{t}:] by Th22;
      reconsider PK = Pr1(X,Y).:K as Subset-Family of X;
      K is open by A4,TOPS_2:11;
      then
A7:   Pr1(X,Y).:K is open by Th19;
      PK is Cover of A by A5,Th21;
      then consider M being Subset-Family of X such that
A8:   M c= Pr1(X,Y).:K and
A9:   M is Cover of A and
A10:  M is finite by A1,A7,COMPTS_1:def 4;
      consider N being Subset-Family of [:X,Y:] such that
A11:  N c= K and
A12:  N is finite and
A13:  Pr1(X,Y).:N = M by A8,A10,Th23;
      reconsider F = Pr1(X,Y).:N as Subset-Family of X;
A14:  N is Cover of [:A,{t}:]
      proof
        let e1,e2 be object;
A15:    A c= union M by A9,SETFAM_1:def 11;
        assume
A16:    [e1,e2] in [:A,{t}:];
        then [e1,e2]`2 in {t} by MCART_1:10;
        then [e1,e2]`2 = t by TARSKI:def 1;
        then
A17:    [e1,e2] = [[e1,e2]`1,t];
        [e1,e2]`1 in A by A16,MCART_1:10;
        then consider X1 being set such that
A18:    [e1,e2]`1 in X1 and
A19:    X1 in M by A15,TARSKI:def 4;
        consider XY being Subset of [:X,Y:] such that
A20:    XY in N and
A21:    Pr1(X,Y).XY = X1 by A13,A19,FUNCT_2:65;
        XY in K by A11,A20;
        then XY in Base-Appr G by A4;
        then consider X2 being Subset of X, Y2 being Subset of Y such that
A22:    XY = [:X2,Y2:] and
        [:X2,Y2:] c= G and
        X2 is open and
        Y2 is open;
        XY meets [:A,{t}:] by A6,A11,A20;
        then consider xy being object such that
A23:    xy in XY and
A24:    xy in [:A,{t}:] by XBOOLE_0:3;
        xy`2 in {t} by A24,MCART_1:10;
        then xy`2 = t by TARSKI:def 1;
        then
A25:    t in Y2 by A22,A23,MCART_1:10;
        XY <> {} by A18,A21,FUNCT_3:8;
        then [e1,e2]`1 in X2 by A18,A21,A22,EQREL_1:50;
        then [e1,e2] in [:X2,Y2:] by A25,A17,ZFMISC_1:87;
        hence [e1,e2] in union N by A20,A22,TARSKI:def 4;
      end;
      then F is Cover of A by Th21;
      then
A26:  A c= union F by SETFAM_1:def 11;
      reconsider H = Pr2(X,Y).:N as Subset-Family of Y;
A27:  now
        let C be set;
        assume C in H;
        then consider D being Subset of [:X,Y:] such that
A28:    D in N and
A29:    C = pr2(the carrier of X, the carrier of Y).:D by Th17;
        D meets [:A,{t}:] by A6,A11,A28;
        then D /\ [:A,{t}:] <> {};
        then consider x being Point of [:X,Y:] such that
A30:    x in D /\ [:A,{t}:];
A31:    x in [:A,{t}:] by A30,XBOOLE_0:def 4;
        then x`1 in A by MCART_1:10;
        then
A32:    (pr2(the carrier of X, the carrier of Y)).(x`1,t) = t by FUNCT_3:def 5;
        x`2 in {t} by A31,MCART_1:10;
        then x`2 = t by TARSKI:def 1;
        then
A33:    x = [x`1,t] by A2,MCART_1:21;
        x in D by A30,XBOOLE_0:def 4;
        hence t in C by A2,A29,A33,A32,FUNCT_2:35;
      end;
      [:A,{t}:] c= union N by A14,SETFAM_1:def 11;
      then N <> {} by A3,ZFMISC_1:2;
      then H <> {} by Th20;
      then
A34:  t in meet H by A27,SETFAM_1:def 1;
A35:  N c= Base-Appr G by A4,A11;
      then
A36:  N is open by TOPS_2:11;
      then meet H is open by A12,Th19,TOPS_2:20;
      then t in Int meet H by A34,TOPS_1:23;
      then reconsider W = meet H as a_neighborhood of t by CONNSP_2:def 1;
      union F is open by A36,Th19,TOPS_2:19;
      then A c= Int union F by A26,TOPS_1:23;
      then reconsider V = union F as a_neighborhood of A by CONNSP_2:def 2;
      take V,W;
      now
        let e;
        assume e in N;
        then e in Base-Appr G by A35;
        then ex X1 being Subset of X, Y1 being Subset of Y st e = [:X1,Y1:] &
        [:X1,Y1:] c= G & X1 is open & Y1 is open;
        hence
        e c= G & ex X1 being Subset of X, Y1 being Subset of Y st e =[:X1
        ,Y1:];
      end;
      hence [:V,W:] c= G by Th15;
    end;
    suppose
      A = {} X;
      then A c= Int {} X;
      then reconsider V = {} X as a_neighborhood of A by CONNSP_2:def 2;
      set W = the a_neighborhood of t;
      take V,W;
      thus [:V,W:] c= G;
    end;
  end;
  hence thesis;
end;
