reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;
reserve e,u for set;

theorem
  succ A is_cofinal_with 1
proof
  deffunc F(set) = A;
  consider psi such that
A1: dom psi = 1 & for B st B in 1 holds psi.B = F(B) from ORDINAL2:sch 3;
  take psi;
  thus dom psi = 1 by A1;
  thus rng psi c= succ A
  proof
    let e be object;
    assume e in rng psi;
    then consider u being object such that
A2: u in 1 and
A3: e = psi.u by A1,FUNCT_1:def 3;
    reconsider u as Ordinal by A2;
    psi.u = A by A1,A2;
    hence thesis by A3,ORDINAL1:6;
  end;
  thus psi is increasing
  by A1,Th14;
A4: psi.{} = A by A1,Lm1,ORDINAL1:6;
  rng psi = {psi.{}} by A1,Lm1,FUNCT_1:4;
  hence thesis by A4,ORDINAL2:23;
end;
