reserve H,S for ZF-formula,
  x for Variable,
  X,Y for set,
  i for Element of NAT,
  e,u for set;
reserve M,M1,M2 for non empty set,
  f for Function,
  v1 for Function of VAR,M1,
  v2 for Function of VAR,M2,
  F,F1,F2 for Subset of WFF,
  W for Universe,
  a,b,c for Ordinal of W,
  A,B,C for Ordinal,
  L for DOMAIN-Sequence of W,
  va for Function of VAR,L.a,
  phi,xi for Ordinal-Sequence of W;
reserve psi for Ordinal-Sequence;

theorem
  A is_cofinal_with succ B implies ex C st A = succ C
proof
  given psi such that
A1: dom psi = succ B and
A2: rng psi c= A and
A3: psi is increasing and
A4: A = sup psi;
  reconsider C = psi.B as Ordinal;
  take C;
  thus A c= succ C
  proof
    let a be Ordinal;
    assume a in A;
    then consider b being Ordinal such that
A5: b in rng psi and
A6: a c= b by A4,ORDINAL2:21;
    consider e being object such that
A7: e in succ B and
A8: b = psi.e by A1,A5,FUNCT_1:def 3;
    reconsider e as Ordinal by A7;
    e c= B by A7,ORDINAL1:22;
    then b c= C by A1,A3,A8,ORDINAL1:6,ORDINAL4:9;
    then b in succ C by ORDINAL1:22;
    hence thesis by A6,ORDINAL1:12;
  end;
  B in succ B by ORDINAL1:6;
  then C in rng psi by A1,FUNCT_1:def 3;
  hence thesis by A2,ORDINAL1:21;
end;
