theorem
  for P1, P2 being Path of a1, b1, Q1, Q2 being Path of b1, c1 st P1, P2
  are_homotopic & Q1, Q2 are_homotopic holds P1 + Q1, P2 + Q2 are_homotopic
proof
  let P1, P2 be Path of a1, b1, Q1, Q2 be Path of b1, c1;
  a1, b1 are_connected & b1, c1 are_connected by BORSUK_2:def 3;
  hence thesis by Th75;
end;
