reserve V for non empty set,
  A,B,A9,B9 for Element of V;
reserve f,f9 for Element of Funcs(V);
reserve m,m1,m2,m3,m9 for Element of Maps V;
reserve a,b for Object of Ens(V);
reserve f,g,f1,f2 for Morphism of Ens(V);

theorem
  (ex x being set st a = {x}) implies a is terminal
proof
  given x being set such that
A1: a = {x};
  let b;
  set h = the Function of @b,@a;
  set m = [[@b,@a],h];
A2: m in Maps(@b,@a) by A1,Th15;
  hence
A3: Hom(b,a)<>{} by Th26;
  m in Hom(b,a) by A2,Th26;
  then reconsider f = m as Morphism of b,a by CAT_1:def 5;
  take f;
  let g be Morphism of b,a;
  reconsider m9 = g as Element of Maps(V);
  g in Hom(b,a) by A3,CAT_1:def 5;
  then
A4: g in Maps(@b,@a) by Th26;
  then
A5: m9 = [[@b,@a],m9`2] by Th16;
  then m9`2 is Function of @b,@a by A4,Lm4;
  hence thesis by A1,A5,FUNCT_2:51;
end;
