reserve B,C,D for Category;

theorem
  for a,b being Object of C
  for f being Morphism of a,b holds f opp is invertible iff f is invertible
proof let a,b be Object of C;
  let f be Morphism of a,b;
  thus f opp is invertible implies f is invertible
  proof assume
A1: Hom(b opp,a opp) <> {} & Hom(a opp,b opp) <> {};
    given gg being Morphism of a opp, b opp such that
A2: (f opp)*gg = id(a opp) & gg*(f opp) = id(b opp);
    thus
A3:   Hom(a,b) <> {} & Hom(b,a) <> {} by A1,Th4;
     reconsider g = opp gg as Morphism of b,a;
    take g;
A4:   g opp = g by Def6,A3
       .= gg by Def7,A1;
    thus f*g =(g opp)*(f opp) by A3,Lm3
       .= id(b opp) by A2,A4
       .= id b by Lm2;
    thus g*f =(f opp)*(g opp) by A3,Lm3
       .= id a by A2,A4,Lm2;
  end;
  assume
A5: Hom(a,b) <> {} & Hom(b,a) <> {};
  given g being Morphism of b,a such that
A6: f*g = id b & g*f = id a;
  thus Hom(b opp,a opp) <> {} & Hom(a opp,b opp) <> {} by A5,Th4;
  take g opp;
  thus (f opp)*(g opp) = g*f by A5,Lm3
         .= id(a opp) by A6,Lm2;
  thus (g opp)*(f opp) = f*g by A5,Lm3
         .= id(b opp) by A6,Lm2;
end;
