theorem
  for A being Path of a1,b1, B being Path of d1,b1, C being Path of b1,
  c1 holds A+-B+B+C, A+C are_homotopic
proof
A1: b1,d1 are_connected by BORSUK_2:def 3;
  a1,b1 are_connected & b1,c1 are_connected by BORSUK_2:def 3;
  hence thesis by A1,Th37;
end;
