reserve C for Simple_closed_curve,
  A,A1,A2 for Subset of TOP-REAL 2,
  p,p1,p2,q ,q1,q2 for Point of TOP-REAL 2,
  n for Element of NAT;

theorem Th8:
  for X,Y being non empty TopSpace, Y0 being non empty SubSpace of
Y, f being Function of X,Y, g being Function of X,Y0 st f = g & f is continuous
  holds g is continuous
proof
  let X,Y be non empty TopSpace, Y0 being non empty SubSpace of Y;
  let f be Function of X,Y, g be Function of X,Y0 such that
A1: f = g and
A2: f is continuous;
  let W being Point of X, G being a_neighborhood of g.W;
  consider V being Subset of Y0 such that
A3: V is open and
A4: V c= G and
A5: g.W in V by CONNSP_2:6;
  g.W in [#]Y0 & [#]Y0 c= [#]Y by PRE_TOPC:def 4;
  then reconsider p = g.W as Point of Y;
  consider C being Subset of Y such that
A6: C is open and
A7: C /\ [#]Y0 = V by A3,TOPS_2:24;
  p in C by A5,A7,XBOOLE_0:def 4;
  then C is a_neighborhood of p by A6,CONNSP_2:3;
  then consider H being a_neighborhood of W such that
A8: f.:H c= C by A1,A2;
  take H;
  g.:H c= V by A1,A7,A8,XBOOLE_1:19;
  hence thesis by A4;
end;
