reserve e for set;

theorem Th18:
  for C being Category holds Alter C is transitive
proof
  let C be Category;
  let o1,o2,o3 be Object of Alter C such that
A1: <^o1,o2^> <> {} & <^o2,o3^> <> {};
  reconsider x1 = o1, x2 = o2, x3 = o3 as Object of C;
A2: <^o1,o3^> = Hom(x1,x3) by Def3;
  <^o1,o2^> = Hom(x1,x2) & <^o2,o3^> = Hom(x2,x3) by Def3;
  hence thesis by A1,A2,CAT_1:24;
end;
