theorem
  for A being Path of a1,b1, B being Path of c1,d1, C being Path of a1,
  c1 holds A+-A+C+B+-B, C are_homotopic
proof
A1: c1,d1 are_connected by BORSUK_2:def 3;
  a1,b1 are_connected & a1,c1 are_connected by BORSUK_2:def 3;
  hence thesis by A1,Th39;
end;
