
theorem Th17:
  for A, B, C being category st A, B are_opposite
  holds A, C are_isomorphic iff B, C are_anti-isomorphic
proof
  let A,B,C be category;
  assume A, B are_opposite;
  then
A1: dualizing-func(A,B) is bijective by Th15;
  hereby
    assume A, C are_isomorphic;
    then consider F being Functor of C,A such that
A2: F is bijective covariant by FUNCTOR0:def 39;
    reconsider F as covariant Functor of C,A by A2;
    dualizing-func(A,B)*F is bijective contravariant by A1,A2,FUNCTOR1:12;
    hence B, C are_anti-isomorphic by FUNCTOR0:def 40;
  end;
  assume B, C are_anti-isomorphic;
  then consider F being Functor of B,C such that
A3: F is bijective contravariant;
  reconsider F as contravariant Functor of B,C by A3;
  F*dualizing-func(A,B) is bijective covariant by A1,A3,FUNCTOR1:12;
  hence thesis;
end;
