reserve a, r, s for Real;

theorem Th13:
  for T being non empty TopSpace holds T is locally_connected iff
  [#]T is locally_connected
proof
  let T be non empty TopSpace;
  T is SubSpace of T by TSEP_1:2;
  then
A1: the TopStruct of T = the TopStruct of (T| [#]T) by PRE_TOPC:8,TSEP_1:5;
  hence T is locally_connected implies [#]T is locally_connected by Th12;
  assume [#]T is locally_connected;
  then T| [#]T is locally_connected;
  hence thesis by A1,Th12;
end;
