reserve x,y for set;

theorem Th48:
  for A being category, B being non empty subcategory of A holds B
  opp is subcategory of A opp
proof
  let A be category, B be non empty subcategory of A;
  reconsider BB = B opp as transitive non empty SubCatStr of A opp by Th47;
A1: A opp, A are_opposite by YELLOW18:def 4;
A2: BB, B are_opposite by YELLOW18:def 4;
  BB is id-inheriting
  proof
    per cases;
    case BB is non empty;
      let o be Object of BB, o9 be Object of A opp;
      reconsider a9 = o9 as Object of A by A1,YELLOW18:def 3;
      reconsider a = o as Object of B by A2,YELLOW18:def 3;
      assume o = o9;
      then idm a9 in <^a,a^> by ALTCAT_2:def 14;
      then idm o9 in <^a,a^> by A1,YELLOW18:10;
      hence thesis by A2,YELLOW18:7;
    end;
    case BB is empty;
    end;
  end;
  hence thesis;
end;
