
theorem
  for C being category, o1,o2,o3 being Object of C st o1,o2 are_iso & o2
  ,o3 are_iso holds o1,o3 are_iso
proof
  let C be category, o1,o2,o3 be Object of C such that
A1: o1,o2 are_iso and
A2: o2,o3 are_iso;
A3: <^o1,o2^> <> {} & <^o2,o3^> <> {} by A1,A2;
  consider B being Morphism of o2,o3 such that
A4: B is iso by A2;
  consider A being Morphism of o1,o2 such that
A5: A is iso by A1;
  <^o2,o1^> <> {} & <^o3,o2^> <> {} by A1,A2;
  hence
A6: <^o1,o3^> <> {} & <^o3,o1^> <> {} by A3,ALTCAT_1:def 2;
  take B * A;
  thus thesis by A3,A6,A5,A4,Th7;
end;
