theorem
  a = {} implies a is initial
proof
  assume
A1: a = {};
  let b;
  Maps(@a,@b) <> {} by A1,Th15;
  hence
A2: Hom(a,b)<>{} by Th26;
  set m = [[@a,@b],{}];
  {} is Function of @a,@b by A1,RELSET_1:12;
  then
 m in Maps(@a,@b) by A1,Th15;
  then m in Hom(a,b) by Th26;
  then reconsider f = m as Morphism of a,b by CAT_1:def 5;
  take f;
  let g be Morphism of a,b;
  reconsider m9 = g as Element of Maps(V);
  g in Hom(a,b) by A2,CAT_1:def 5;
  then
A3: g in Maps(@a,@b) by Th26;
  then
A4: m9 = [[@a,@b],m9`2] by Th16;
  then m9`2 is Function of @a,@b by A3,Lm4;
  hence thesis by A1,A4;
end;
