reserve e for set;
reserve C,C1,C2,C3 for AltCatStr;
reserve C for non empty AltCatStr,
  o for Object of C;
reserve C for non empty transitive AltCatStr;

theorem
  for C being category, D being non empty subcategory of C for o being
  Object of D, o9 being Object of C st o = o9 holds idm o = idm o9
proof
  let C be category, D be non empty subcategory of C;
  let o be Object of D, o9 be Object of C;
  assume
A1: o = o9;
  then reconsider m = idm o9 as Morphism of o,o by Def14;
A2: idm o9 in <^o,o^> by A1,Def14;
  now
    let p be Object of D such that
A3: <^o,p^> <> {};
    reconsider p9 = p as Object of C by Th29;
A4: <^o9,p9^> <> {} by A1,A3,Th31,XBOOLE_1:3;
    let a be Morphism of o,p;
    reconsider n = a as Morphism of o9,p9 by A1,A3,Th33;
    thus a*m = n*(idm o9) by A1,A2,A3,Th32
      .= a by A4,ALTCAT_1:def 17;
  end;
  hence thesis by ALTCAT_1:def 17;
end;
