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 Th13:
  [s1,t1],[s2,t2] are_connected implies for L being Path of [s1,t1
  ],[s2,t2] holds pr1 L is Path of s1,s2
proof
  assume
A1: [s1,t1], [s2,t2] are_connected;
  let L be Path of [s1,t1],[s2,t2];
  set f = pr1 L;
A2: dom f = I & dom f = dom L by FUNCT_2:def 1,MCART_1:def 12;
  then j0 in dom L;
  then
A3: f.0 = (L.0)`1 by MCART_1:def 12
    .= [s1,t1]`1 by A1,BORSUK_2:def 2
    .= s1;
  j1 in dom L by A2;
  then
A4: f.1 = (L.1)`1 by MCART_1:def 12
    .= [s2,t2]`1 by A1,BORSUK_2:def 2
    .= s2;
  L is continuous by A1,BORSUK_2:def 2;
  then
A5: f is continuous by Th9;
  then s1,s2 are_connected by A3,A4;
  hence thesis by A5,A3,A4,BORSUK_2:def 2;
end;
