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

theorem Th17:
  X is locally_connected implies for A being non empty Subset of X
  st A is open holds A is locally_connected
proof
  assume
A1: X is locally_connected;
  let A be non empty Subset of X such that
A2: A is open;
  for x being Point of X|A holds X|A is_locally_connected_in x
  proof
    let x be Point of X|A;
    x in [#](X|A);
    then
A3: x in A by PRE_TOPC:def 5;
    then reconsider x1=x as Point of X;
    X is_locally_connected_in x1 by A1;
    then A is_locally_connected_in x1 by A2,A3,Th16;
    then
    ex x2 being Point of X|A st x2=x1 & X|A is_locally_connected_in x2;
    hence thesis;
  end;
  then X|A is locally_connected;
  hence thesis;
end;
