reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem
  X is locally_connected & X is normal implies for A,B being Subset of X
  st A <> {} & B <> {} & A is closed & B is closed & A misses B holds (A is
connected & B is connected implies ex R,S being Subset of X st R is connected &
  S is connected & R is open & S is open & A c= R & B c= S & R misses S)
proof
  assume that
A1: X is locally_connected and
A2: X is normal;
  let A,B be Subset of X such that
A3: A <> {} and
A4: B <> {} and
A5: A is closed and
A6: B is closed & A misses B;
  B = [#] X \ ([#] X \ B) by PRE_TOPC:3;
  then
A7: [#] X \ B <> [#] X by A4,PRE_TOPC:4;
  A <> {} X by A3;
  then consider x being Point of X such that
A8: x in A by PRE_TOPC:12;
  A c= B` & B` is open by A6,SUBSET_1:23;
  then consider G being Subset of X such that
A9: G is open and
A10: A c= G and
A11: Cl G c= B` by A2,A3,A5,A7,Th20;
A12: Cl G misses B by A11,SUBSET_1:23;
A13: x in G by A10,A8;
  reconsider G as non empty Subset of X by A3,A10;
  x in [#](X|G) by A13,PRE_TOPC:def 5;
  then reconsider y=x as Point of X|G;
A14: Cl G misses (Cl G)` by XBOOLE_1:79;
  assume that
A15: A is connected and
A16: B is connected;
  set H=Component_of y;
  reconsider H1=H as Subset of X by PRE_TOPC:11;
  take R=H1;
A17: H is a_component by CONNSP_1:40;
  then
A18: H1 is_a_component_of G by CONNSP_1:def 6;
  A c= [#](X|G) by A10,PRE_TOPC:def 5;
  then reconsider A1=A as Subset of X|G;
A19: H is connected & y in H by CONNSP_1:38;
  A1 is connected by A15,CONNSP_1:23;
  then A1 misses H or A1 c= H by A17,CONNSP_1:36;
  then
A20: A1 /\ H = {} or A1 c= H by XBOOLE_0:def 7;
  H c= [#](X|G);
  then
A21: R c= G by PRE_TOPC:def 5;
  G c= Cl G by PRE_TOPC:18;
  then
A22: R c= Cl G by A21;
  B <> {} X by A4;
  then consider z being Point of X such that
A23: z in B by PRE_TOPC:12;
A24: B c= (Cl G)` by A12,SUBSET_1:23;
  then reconsider C = (Cl G)` as non empty Subset of X by A23;
  z in (Cl G)` by A23,A24;
  then z in [#](X|C) by PRE_TOPC:def 5;
  then reconsider z1=z as Point of X|C;
  set V=Component_of z1;
  reconsider V1=V as Subset of X by PRE_TOPC:11;
  take S=V1;
A25: V is a_component by CONNSP_1:40;
  B c= [#](X|(Cl G)`) by A24,PRE_TOPC:def 5;
  then reconsider B1=B as Subset of X|(Cl G)`;
A26: V is connected & z1 in V by CONNSP_1:38;
  B1 is connected by A16,CONNSP_1:23;
  then B1 misses V or B1 c= V by A25,CONNSP_1:36;
  then
A27: B1 /\ V = {} or B1 c= V by XBOOLE_0:def 7;
  V c= [#](X|(Cl G)`);
  then S c= (Cl G)` by PRE_TOPC:def 5;
  then R /\ S c= Cl G /\ (Cl G)` by A22,XBOOLE_1:27;
  then R /\ S c= {} X by A14,XBOOLE_0:def 7;
  then
A28: R /\ S = {};
  V1 is_a_component_of (Cl G)` by A25,CONNSP_1:def 6;
  hence thesis by A1,A9,A8,A18,A20,A19,A23,A27,A26,A28,Th18,CONNSP_1:23
,XBOOLE_0:def 4,def 7;
end;
