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;
