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 Th12:
  [s1,t1],[s2,t2] are_connected implies t1,t2 are_connected
proof
  given L being Function of I[01], [:S,T:] such that
A1: L is continuous and
A2: L.0 = [s1,t1] and
A3: L.1 = [s2,t2];
  take f = pr2 L;
  thus f is continuous by A1,Th10;
A4: dom f = I & dom f = dom L by FUNCT_2:def 1,MCART_1:def 13;
  then j0 in dom L;
  hence f.0 = [s1,t1]`2 by A2,MCART_1:def 13
    .= t1;
  j1 in dom L by A4;
  hence f.1 = [s2,t2]`2 by A3,MCART_1:def 13
    .= t2;
end;
