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;
