theorem
  for A being Path of a1,b1, B being Path of b1,c1, C being Path of c1,
d1, D being Path of d1,e1, E being Path of f1,c1 holds A+(B+C)+D, A+B+-E+(E+C+D
  ) are_homotopic
proof
A1: c1,d1 are_connected & d1,e1 are_connected by BORSUK_2:def 3;
A2: a1,f1 are_connected by BORSUK_2:def 3;
  a1,b1 are_connected & b1,c1 are_connected by BORSUK_2:def 3;
  hence thesis by A1,A2,Th43;
end;
