reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;

theorem Th9:
  ex M st M c= card A & A is_cofinal_with M & for B st A
  is_cofinal_with B holds M c= B
proof
  defpred P[Ordinal] means $1 c= card A & A is_cofinal_with $1;
A1: ex B st P[B] by Th8;
  consider C such that
A2: P[C] and
A3: for B st P[B] holds C c= B from ORDINAL1:sch 1(A1);
  take M = card C;
  consider B such that
A4: B c= M and
A5: C is_cofinal_with B by Th8;
A6: M c= C by CARD_1:8;
  then
A7: B c= C by A4;
  then B c= card A by A2,XBOOLE_1:1;
  then C c= B by A2,A3,A5,ORDINAL4:37;
  then
A8: C = B by A7;
  hence M c= card A & A is_cofinal_with M by A2,A4,A6,XBOOLE_0:def 10;
  let B;
  assume that
A9: A is_cofinal_with B and
A10: not M c= B;
A11: C = M by A4,A6,A8;
  then not B c= card A by A3,A9,A10;
  hence contradiction by A2,A11,A10,XBOOLE_1:1;
end;
