
theorem Th23:
  for S, T being non empty TopSpace, f being continuous Function
of S,T, a, b being Point of S st a,b are_connected holds f.a,f.b are_connected
proof
  let S, T be non empty TopSpace;
  let f be continuous Function of S,T;
  let a, b be Point of S;
  given g being Function of I[01], S such that
A1: g is continuous and
A2: g.0 = a and
A3: g.1 = b;
  take h = f*g;
  thus h is continuous by A1;
  thus h.0 = f.(g.j0) by FUNCT_2:15
    .= f.a by A2;
  thus h.1 = f.(g.j1) by FUNCT_2:15
    .= f.b by A3;
end;
