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

theorem Th14:
  for x holds X is_locally_connected_in x iff for U1 being non
empty Subset of X st U1 is open & x in U1 ex x1 being Point of X|U1 st x1=x & x
  in Int(Component_of x1)
proof
  let x;
A1: X is_locally_connected_in x implies for U1 being non empty Subset of X
  st U1 is open & x in U1 ex x1 being Point of X|U1 st x1=x & x in Int(
  Component_of x1)
  proof
    assume
A2: X is_locally_connected_in x;
    let U1 be non empty Subset of X such that
A3: U1 is open and
A4: x in U1;
    x in [#](X|U1) by A4,PRE_TOPC:def 5;
    then reconsider x1=x as Point of X|U1;
    reconsider S1=Component_of x1 as Subset of X|U1;
    take x1;
    reconsider S=S1 as Subset of X by PRE_TOPC:11;
A5: x in S by CONNSP_1:38;
    S1 is a_component by CONNSP_1:40;
    then
A6: S is_a_component_of U1 by CONNSP_1:def 6;
    U1 is a_neighborhood of x by A3,A4,Th3;
    then S is a_neighborhood of x by A2,A6,A5,Th13;
    then S1 is a_neighborhood of x1 by Th10;
    hence thesis by Def1;
  end;
  (for U1 being non empty Subset of X st U1 is open & x in U1 ex x1 being
  Point of X|U1 st x1=x & x in Int(Component_of x1)) implies X
  is_locally_connected_in x
  proof
    assume
A7: for U1 being non empty Subset of X st U1 is open & x in U1 ex x1
    being Point of X|U1 st x1=x & x in Int(Component_of x1);
    for U1 being Subset of X st U1 is a_neighborhood of x ex V1 being
    Subset of X st V1 is a_neighborhood of x & V1 is connected & V1 c= U1
    proof
      let U1 be Subset of X;
      assume U1 is a_neighborhood of x;
      then consider V being non empty Subset of X such that
A8:   V is a_neighborhood of x and
A9:   V is open and
A10:  V c= U1 by Th5;
      consider x1 being Point of X|V such that
A11:  x1=x and
A12:  x in Int(Component_of x1) by A7,A8,A9,Th4;
      set S1=Component_of x1;
      reconsider S=S1 as Subset of X by PRE_TOPC:11;
      take S;
      S1 c= [#](X|V);
      then
A13:  S1 is connected & S c= V by PRE_TOPC:def 5;
      S1 is a_neighborhood of x1 by A11,A12,Def1;
      hence thesis by A9,A10,A11,A13,Th9,CONNSP_1:23;
    end;
    hence thesis;
  end;
  hence thesis by A1;
end;
