theorem
  for A1,A2 being Path of a1,b1, B being Path of b1,c1 holds A1,A2
  are_homotopic implies A1, A2+B+-B are_homotopic
proof
  let A1,A2 be Path of a1,b1;
  a1,b1 are_connected & b1,c1 are_connected by BORSUK_2:def 3;
  hence thesis by Th19;
end;
