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

theorem
  X is locally_connected implies for E being non empty Subset of X, C
  being non empty Subset of X|E st C is connected & C is open ex H being Subset
  of X st H is open & H is connected & C = E /\ H
proof
  assume
A1: X is locally_connected;
  let E be non empty Subset of X, C be non empty Subset of X|E such that
A2: C is connected and
A3: C is open;
  C in the topology of X|E by A3,PRE_TOPC:def 2;
  then consider G being Subset of X such that
A4: G in the topology of X and
A5: C = G /\ [#](X|E) by PRE_TOPC:def 4;
A6: C = G /\ E by A5,PRE_TOPC:def 5;
  reconsider G as non empty Subset of X by A5;
A7: G is open by A4,PRE_TOPC:def 2;
  reconsider C1=C as Subset of X by PRE_TOPC:11;
  C <> {} (X|E);
  then consider x being Point of X|E such that
A8: x in C by PRE_TOPC:12;
  x in G by A5,A8,XBOOLE_0:def 4;
  then x in [#](X|G) by PRE_TOPC:def 5;
  then reconsider y=x as Point of X|G;
  set H1 = Component_of y;
  reconsider H=H1 as Subset of X by PRE_TOPC:11;
  take H;
A9: H1 is a_component by CONNSP_1:40;
  then H is_a_component_of G by CONNSP_1:def 6;
  hence H is open by A1,A7,Th18;
  C c= G by A5,XBOOLE_1:17;
  then C c= [#](X|G) by PRE_TOPC:def 5;
  then reconsider C2=C1 as Subset of X|G;
  H1 c= [#](X|G);
  then
A10: H c= G by PRE_TOPC:def 5;
  C1 is connected by A2,CONNSP_1:23;
  then C2 is connected by CONNSP_1:23;
  then C2 misses H or C2 c= H by A9,CONNSP_1:36;
  then
A11: C2 /\ H = {}(X|G) or C2 c= H by XBOOLE_0:def 7;
A12: x in H1 by CONNSP_1:38;
  H /\ (G /\ E) c= C by A6,XBOOLE_0:def 4;
  then (H /\ G) /\ E c= C by XBOOLE_1:16;
  then
A13: E /\ H c= C by A10,XBOOLE_1:28;
  thus H is connected by CONNSP_1:23;
  C c= E by A6,XBOOLE_1:17;
  then C c= E /\ H by A8,A11,A12,XBOOLE_0:def 4;
  hence thesis by A13,XBOOLE_0:def 10;
end;
