reserve a, r, s for Real;

theorem Th12:
  for S, T being non empty TopSpace st the TopStruct of S = the
  TopStruct of T & S is locally_connected holds T is locally_connected
proof
  let S, T be non empty TopSpace such that
A1: the TopStruct of S = the TopStruct of T and
A2: S is locally_connected;
  let t be Point of T;
  reconsider s = t as Point of S by A1;
  let U be Subset of T;
  reconsider U1 = U as Subset of S by A1;
  assume U is a_neighborhood of t;
  then
A3: U1 is a_neighborhood of s by A1,Th8;
  S is_locally_connected_in s by A2;
  then consider V1 being Subset of S such that
A4: V1 is a_neighborhood of s and
A5: V1 is connected and
A6: V1 c= U1 by A3;
  reconsider V = V1 as Subset of T by A1;
  take V;
  thus V is a_neighborhood of t by A1,A4,Th8;
  thus V is connected by A1,A5,Th7;
  thus thesis by A6;
end;
