reserve S, T, Y for non empty TopSpace,
  s, s1, s2, s3 for Point of S,
  t, t1, t2, t3 for Point of T,
  l1, l2 for Path of [s1,t1],[s2,t2],
  H for Homotopy of l1 ,l2;

theorem Th15:
  s1,s2 are_connected & t1,t2 are_connected implies [s1,t1],[s2,t2
  ] are_connected
proof
  given f being Function of I[01], S such that
A1: f is continuous and
A2: f.0 = s1 and
A3: f.1 = s2;
  given g being Function of I[01], T such that
A4: g is continuous and
A5: g.0 = t1 and
A6: g.1 = t2;
  take <:f,g:>;
  thus <:f,g:> is continuous by A1,A4,YELLOW12:41;
A7: dom f = I & dom g = I by FUNCT_2:def 1;
  hence <:f,g:>.0 = [f.j0,g.j0] by FUNCT_3:49
    .= [s1,t1] by A2,A5;
  thus <:f,g:>.1 = [f.j1,g.j1] by A7,FUNCT_3:49
    .= [s2,t2] by A3,A6;
end;
