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 Th20:
  A is_cofinal_with B implies B c= A
proof
  given psi such that
A1: dom psi = B and
A2: rng psi c= A and
A3: psi is increasing and
  A = sup psi;
  let C;
  assume C in B;
  then C c= psi.C & psi.C in rng psi by A1,A3,FUNCT_1:def 3,ORDINAL4:10;
  hence thesis by A2,ORDINAL1:12;
end;
