
theorem
  for C being strict Category holds (C opp) opp = C
  proof
    let C be strict Category;
    C opp is Category;
    then reconsider Copp = CatStr (# the carrier of C, the carrier' of C,
                                      the Target of C, the Source of C,
                                      ~the Comp of C #) as
      strict Category by OPPCAT_1:def 1;
    C opp = Copp by OPPCAT_1:def 1;
    then
A1: (C opp) opp = CatStr (#the carrier of Copp, the carrier' of Copp,
                             the Target of Copp, the Source of Copp,
                             ~ the Comp of Copp#)
      by OPPCAT_1:def 1;
    proj1 the Comp of C c= proj1 [: [:the carrier' of C,the carrier' of C:] ,
    the carrier' of C :] by XTUPLE_0:8;
    hence thesis by A1,FUNCT_4:52;
  end;
