reserve x,y for set;

theorem
  for A being category, B being non empty subcategory of A holds B,B opp
  are_anti-isomorphic_under dualizing-func(A, A opp)
proof
  let A be category, B be non empty subcategory of A;
  set F = dualizing-func(A, A opp);
A1: B, B opp are_opposite by YELLOW18:def 4;
  thus B is subcategory of A & B opp is subcategory of A opp by Th48;
  take G = dualizing-func(B, B opp);
  thus G is bijective;
A2: A, A opp are_opposite by YELLOW18:def 4;
  hereby
    let a be Object of B, a1 be Object of A;
    assume a = a1;
    hence G.a = a1 by A1,YELLOW18:def 5
      .= F.a1 by A2,YELLOW18:def 5;
  end;
  let b,c be Object of B, b1,c1 be Object of A such that
A3: <^b,c^> <> {} and
A4: b = b1 & c = c1;
  let f be Morphism of b,c, f1 be Morphism of b1,c1 such that
A5: f = f1;
A6: <^b,c^> c= <^b1,c1^> & f in <^b,c^> by A3,A4,ALTCAT_2:31;
  then
A7: <^F.c1,F.b1^> <> {} by FUNCTOR0:def 19;
  thus G.f = f by A1,A3,YELLOW18:def 5
    .= F.f1 by A2,A5,A6,YELLOW18:def 5
    .= Morph-Map(F,b1,c1).f1 by A6,A7,FUNCTOR0:def 16;
end;
