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

theorem Th16:
  X is_locally_connected_in x implies for A being non empty Subset
  of X st A is open & x in A holds A is_locally_connected_in x
proof
  assume
A1: X is_locally_connected_in x;
  let A be non empty Subset of X such that
A2: A is open and
A3: x in A;
  reconsider B = A as non empty Subset of X;
A4: [#](X|A) = A by PRE_TOPC:def 5;
  for C being non empty Subset of X st B = C ex x1 being Point of X|C st
  x1=x & X|C is_locally_connected_in x1
  proof
    let C be non empty Subset of X;
    assume
A5: B = C;
    then reconsider y=x as Point of X|C by A3,A4;
    take x1=y;
    for U1 being Subset of X|B st U1 is a_neighborhood of x1 ex V being
    Subset of X|B st V is a_neighborhood of x1 & V is connected & V c= U1
    proof
      let U1 be Subset of X|B such that
A6:   U1 is a_neighborhood of x1;
      reconsider U2=U1 as Subset of X by PRE_TOPC:11;
      U2 is a_neighborhood of x by A2,A5,A6,Th9;
      then consider V being Subset of X such that
A7:   V is a_neighborhood of x & V is connected and
A8:   V c= U2 by A1;
      reconsider V1= V as Subset of X|B by A8,XBOOLE_1:1;
      take V1;
      thus thesis by A5,A7,A8,Th10,CONNSP_1:23;
    end;
    hence thesis by A5;
  end;
  hence thesis;
end;
