reserve phi,fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  f,g for Function,
  X for set,
  x,y,z for object;
reserve f1,f2 for Ordinal-Sequence;
reserve W for Universe;
reserve A1,B1 for Ordinal of W,
  phi for Ordinal-Sequence of W;
reserve L for Sequence;
reserve e,u for set;

theorem
  A is_cofinal_with B implies (A is limit_ordinal iff B is limit_ordinal )
proof
  given psi such that
A1: dom psi = B and
A2: rng psi c= A and
A3: psi is increasing and
A4: A = sup psi;
  thus A is limit_ordinal implies B is limit_ordinal
  proof
    assume
A5: A is limit_ordinal;
    now
      let C;
      reconsider c = psi.C as Ordinal;
      assume
A6:   C in B;
      then psi.C in rng psi by A1,FUNCT_1:def 3;
      then succ c in A by A2,A5,ORDINAL1:28;
      then consider b being Ordinal such that
A7:   b in rng psi and
A8:   succ c c= b by A4,ORDINAL2:21;
      consider e being object such that
A9:   e in B and
A10:  b = psi.e by A1,A7,FUNCT_1:def 3;
      reconsider e as Ordinal by A9;
      c in succ c by ORDINAL1:6;
      then not e c= C by A1,A3,A6,A8,A10,Th9,ORDINAL1:5;
      then C in e by ORDINAL1:16;
      then succ C c= e by ORDINAL1:21;
      hence succ C in B by A9,ORDINAL1:12;
    end;
    hence thesis by ORDINAL1:28;
  end;
  thus thesis by A1,A3,A4,Th16;
end;
