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
  dom psi <> {} & dom psi is limit_ordinal & psi is increasing & A
  is_limes_of psi implies A is_cofinal_with dom psi
proof
  assume that
A1: dom psi <> {} & dom psi is limit_ordinal and
A2: psi is increasing and
A3: A is_limes_of psi;
  take psi;
  thus dom psi = dom psi;
  sup psi = lim psi & A = lim psi by A1,A2,A3,ORDINAL2:def 10,ORDINAL4:8;
  hence thesis by A2,ORDINAL2:49;
end;
