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: a1,c1 are_connected by BORSUK_2:def 3;
  a1,b1 are_connected & d1,c1 are_connected by BORSUK_2:def 3;
  hence thesis by A1,Th41;
end;
