
theorem Th4:
  for A,B,C being category st A,B are_equivalent & B,C are_equivalent
  holds A,C are_equivalent
proof
  let A,B,C be category;
  given F1 being covariant Functor of A,B,
  G1 being covariant Functor of B,A such that
A1: G1*F1, id A are_naturally_equivalent and
A2: F1*G1, id B are_naturally_equivalent;
  given F2 being covariant Functor of B,C,
  G2 being covariant Functor of C,B such that
A3: G2*F2, id B are_naturally_equivalent and
A4: F2*G2, id C are_naturally_equivalent;
  take F = F2*F1, G = G1*G2;
  the FunctorStr of F1 = the FunctorStr of F1;
  then
A5: (the FunctorStr of G1)*F1 = G1*F1 by Th3;
  the FunctorStr of G2 = the FunctorStr of G2;
  then
A6: (the FunctorStr of F2)*G2 = F2*G2 by Th3;
A7: G1* id B = the FunctorStr of G1 by FUNCTOR3:5;
A8: F2* id B = the FunctorStr of F2 by FUNCTOR3:5;
A9: G*F2 = G1*(G2*F2) by FUNCTOR0:32;
A10: F*G1 = F2*(F1*G1) by FUNCTOR0:32;
A11: G*F2*F1 = G*F by FUNCTOR0:32;
A12: F*G1*G2 = F*G by FUNCTOR0:32;
A13: G*F2, G1* id B are_naturally_equivalent by A3,A9,FUNCTOR3:35;
A14: F*G1, F2* id B are_naturally_equivalent by A2,A10,FUNCTOR3:35;
A15: G*F, G1*F1 are_naturally_equivalent by A5,A7,A11,A13,FUNCTOR3:36;
  F*G, F2*G2 are_naturally_equivalent by A6,A8,A12,A14,FUNCTOR3:36;
  hence thesis by A1,A4,A15,FUNCTOR3:33;
end;
