reserve p, q, x, y for Real,
  n for Nat;
reserve X for non empty TopSpace,
  a, b, c, d, e, f for Point of X,
  T for non empty pathwise_connected TopSpace,
  a1, b1, c1, d1, e1, f1 for Point of T;

theorem
  for A1,A2 being Path of a1,b1, B being Path of c1,b1 holds A1,A2
  are_homotopic implies A1, A2+-B+B are_homotopic
proof
  a1,b1 are_connected & c1,b1 are_connected by BORSUK_2:def 3;
  hence thesis by Th21;
end;
