reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;
reserve o,m for set;
reserve F,F1,F2,F3 for Functor of A,B,
  G,G1,G2,G3 for Functor of B,C,
  H,H1,H2 for Functor of C,D,
  s for natural_transformation of F1,F2,
  s9 for natural_transformation of F2,F3,
  t for natural_transformation of G1,G2,
  t9 for natural_transformation of G2,G3,
  u for natural_transformation of H1,H2;

theorem Th44:
  A,B are_equivalent & B,C are_equivalent implies A,C are_equivalent
proof
  given F1 being Functor of A,B, G1 being Functor of B,A such that
A1: G1*F1 ~= id A and
A2: F1*G1 ~= id B;
  given F2 being Functor of B,C, G2 being Functor of C,B such that
A3: G2*F2 ~= id B and
A4: F2*G2 ~= id C;
  take F2*F1,G1*G2;
  (G1*G2)*F2 = G1*(G2*F2) by RELAT_1:36;
  then
A5: (G1*G2)*F2 ~= G1 by A3,Th42;
  (G1*G2)*(F2*F1) = ((G1*G2)*F2)*F1 by RELAT_1:36;
  then (G1*G2)*(F2*F1) ~= G1*F1 by A5,Th41;
  hence (G1*G2)*(F2*F1) ~= id A by A1,NATTRA_1:29;
  (F2*F1)*G1 = F2*(F1*G1) by RELAT_1:36;
  then
A6: (F2*F1)*G1 ~= F2 by A2,Th42;
  (F2*F1)*(G1*G2) = ((F2*F1)*G1)*G2 by RELAT_1:36;
  then (F2*F1)*(G1*G2) ~= F2*G2 by A6,Th41;
  hence thesis by A4,NATTRA_1:29;
end;
