
theorem Th14:
  for S, T being TopSpace, A being Subset of S, B being Subset of
  T holds Cl [:A,B:] = [:Cl A,Cl B:]
proof
  let S, T be TopSpace, A be Subset of S, B be Subset of T;
  hereby
    let x be object;
    assume
A1: x in Cl [:A,B:];
    then reconsider S1 = S, T1 = T as non empty TopSpace;
    reconsider p = x as Point of [:S1,T1:] by A1;
    consider K being Basis of p such that
A2: for Q being Subset of [:S1,T1:] st Q in K holds [:A,B:] meets Q by A1,
YELLOW13:17;
    consider p1 being Point of S1, p2 being Point of T1 such that
A3: p = [p1,p2] by BORSUK_1:10;
    for G being Subset of T1 st G is open & p2 in G holds B meets G
    proof
      let G be Subset of T1;
      assume G is open & p2 in G;
      then [p1,p2] in [:[#]S1,G:] & [:[#]S1,G:] is open by BORSUK_1:6
,ZFMISC_1:87;
      then consider V being Subset of [:S1,T1:] such that
A4:   V in K and
A5:   V c= [:[#]S1,G:] by A3,YELLOW_8:def 1;
      [:A,B:] meets V by A2,A4;
      then consider a being object such that
A6:   a in [:A,B:] & a in V by XBOOLE_0:3;
      a in [:A,B:] /\ [:[#]S1,G:] by A5,A6,XBOOLE_0:def 4;
      then a in [:A/\[#]S1,B/\G:] by ZFMISC_1:100;
      then
      ex a1, a2 being object st a1 in A/\[#]S1 & a2 in B/\G & a = [a1,a2] by
ZFMISC_1:def 2;
      hence thesis;
    end;
    then
A7: p2 in Cl B by PRE_TOPC:def 7;
    for G being Subset of S1 st G is open & p1 in G holds A meets G
    proof
      let G be Subset of S1;
      assume G is open & p1 in G;
      then [p1,p2] in [:G,[#]T1:] & [:G,[#]T1:] is open by BORSUK_1:6
,ZFMISC_1:87;
      then consider V being Subset of [:S1,T1:] such that
A8:   V in K and
A9:   V c= [:G,[#]T1:] by A3,YELLOW_8:def 1;
      [:A,B:] meets V by A2,A8;
      then consider a being object such that
A10:  a in [:A,B:] & a in V by XBOOLE_0:3;
      a in [:A,B:] /\ [:G,[#]T1:] by A9,A10,XBOOLE_0:def 4;
      then a in [:A/\G,B/\[#]T1:] by ZFMISC_1:100;
      then
      ex a1, a2 being object st a1 in A/\G & a2 in B/\[#]T1 & a = [a1,a2] by
ZFMISC_1:def 2;
      hence thesis;
    end;
    then p1 in Cl A by PRE_TOPC:def 7;
    hence x in [:Cl A,Cl B:] by A3,A7,ZFMISC_1:87;
  end;
  let x be object;
  assume x in [:Cl A,Cl B:];
  then consider x1, x2 being object such that
A11: x1 in Cl A and
A12: x2 in Cl B and
A13: x = [x1,x2] by ZFMISC_1:def 2;
  reconsider S1 = S, T1 = T as non empty TopSpace by A11,A12;
  reconsider x1 as Point of S1 by A11;
  consider K1 being Basis of x1 such that
A14: for Q being Subset of S1 st Q in K1 holds A meets Q by A11,YELLOW13:17;
  reconsider x2 as Point of T1 by A12;
  consider K2 being Basis of x2 such that
A15: for Q being Subset of T1 st Q in K2 holds B meets Q by A12,YELLOW13:17;
  for G being Subset of [:S1,T1:] st G is open & [x1,x2] in G holds [:A,B
  :] meets G
  proof
    let G be Subset of [:S1,T1:] such that
A16: G is open and
A17: [x1,x2] in G;
    consider F being Subset-Family of [:S1,T1:] such that
A18: G = union F and
A19: for e being set st e in F ex X1 being Subset of S1, Y1 being
    Subset of T1 st e = [:X1,Y1:] & X1 is open & Y1 is open by A16,BORSUK_1:5;
    consider Z being set such that
A20: [x1,x2] in Z and
A21: Z in F by A17,A18,TARSKI:def 4;
    consider X1 being Subset of S1, Y1 being Subset of T1 such that
A22: Z = [:X1,Y1:] and
A23: X1 is open and
A24: Y1 is open by A19,A21;
    x2 in Y1 by A20,A22,ZFMISC_1:87;
    then consider V2 being Subset of T1 such that
A25: V2 in K2 and
A26: V2 c= Y1 by A24,YELLOW_8:def 1;
    B meets V2 by A15,A25;
    then consider a2 being object such that
A27: a2 in B and
A28: a2 in V2 by XBOOLE_0:3;
    x1 in X1 by A20,A22,ZFMISC_1:87;
    then consider V1 being Subset of S1 such that
A29: V1 in K1 and
A30: V1 c= X1 by A23,YELLOW_8:def 1;
    A meets V1 by A14,A29;
    then consider a1 being object such that
A31: a1 in A and
A32: a1 in V1 by XBOOLE_0:3;
    [a1,a2] in Z by A22,A30,A32,A26,A28,ZFMISC_1:87;
    then
A33: [a1,a2] in union F by A21,TARSKI:def 4;
    [a1,a2] in [:A,B:] by A31,A27,ZFMISC_1:87;
    hence thesis by A18,A33,XBOOLE_0:3;
  end;
  hence thesis by A13,PRE_TOPC:def 7;
end;
