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 Th23:
  for l1 being Path of [s1,t1],[s2,t2], l2 being Path of [s2,t2],[
  s3,t3], p1 being Path of s1,s2, p2 being Path of s2,s3 st [s1,t1],[s2,t2]
are_connected & [s2,t2],[s3,t3] are_connected & p1 = pr1 l1 & p2 = pr1 l2 holds
  pr1 (l1+l2) = p1 + p2
proof
  let l1 be Path of [s1,t1],[s2,t2], l2 be Path of [s2,t2],[s3,t3], p1 be Path
  of s1,s2, p2 be Path of s2,s3 such that
A1: [s1,t1],[s2,t2] are_connected & [s2,t2],[s3,t3] are_connected and
A2: p1 = pr1 l1 and
A3: p2 = pr1 l2;
A4: s1,s2 are_connected & s2,s3 are_connected by A1,Th11;
  now
A5: dom l2 = I by FUNCT_2:def 1;
A6: dom l1 = I by FUNCT_2:def 1;
    let x be Element of I[01];
A7: dom (l1+l2) = I by FUNCT_2:def 1;
    per cases;
    suppose
A8:   x <= 1/2;
      then
A9:   2*x is Point of I[01] by BORSUK_6:3;
      thus (pr1 (l1+l2)).x = ((l1+l2).x)`1 by A7,MCART_1:def 12
        .= (l1.(2*x))`1 by A1,A8,BORSUK_2:def 5
        .= p1.(2*x) by A2,A6,A9,MCART_1:def 12
        .= (p1 + p2).x by A4,A8,BORSUK_2:def 5;
    end;
    suppose
A10:  x >= 1/2;
      then
A11:  2*x-1 is Point of I[01] by BORSUK_6:4;
      thus (pr1 (l1+l2)).x = ((l1+l2).x)`1 by A7,MCART_1:def 12
        .= (l2.(2*x-1))`1 by A1,A10,BORSUK_2:def 5
        .= p2.(2*x-1) by A3,A5,A11,MCART_1:def 12
        .= (p1 + p2).x by A4,A10,BORSUK_2:def 5;
    end;
  end;
  hence thesis;
end;
