reserve i,j,k,x for object;

theorem
  for C being with_units non empty AltCatStr for o1,o2 being Object of
  C st <^o1,o2^> <> {} for a being Morphism of o1,o2 holds (idm o2)*a = a
proof
  let C be with_units non empty AltCatStr;
  let o1,o2 be Object of C such that
A1: <^o1,o2^> <> {};
  let a be Morphism of o1,o2;
  the Comp of C is with_left_units by Def16;
  then consider d being set such that
A2: d in (the Arrows of C).(o2,o2) and
A3: for o9 being Element of C, f be set st f in (the Arrows of C).(o9,o2
  ) holds (the Comp of C).(o9,o2,o2).(d,f) = f;
  reconsider d as Morphism of o2,o2 by A2;
  idm o2 in <^o2,o2^> by Th13;
  then d = d*idm o2 by Def17
    .= (the Comp of C).(o2,o2,o2).(d,idm o2) by A2,Def8
    .= idm o2 by A3,Th13;
  hence (idm o2)*a = (the Comp of C).(o1,o2,o2).(d,a) by A1,A2,Def8
    .= a by A1,A3;
end;
