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;
reserve a,b for Aleph;

theorem
  cf 0 = 0 & cf card (n+1) = 1
proof
A1: succ n is_cofinal_with 1 by ORDINAL3:73;
  card (n+1) = n+1 & Segm(n+1) = succ Segm n by NAT_1:38;
  then cf card (n+1) c= 1 by A1,Def1;
  then
A2: cf card (n+1) = 1 or cf card (n+1) = 0 & {} c= n & n in succ n by
ORDINAL1:6,ORDINAL3:16;
  cf 0 c= 0 & 0 c= cf 0 by Def1;
  hence cf 0 = 0;
  card (n+1) is_cofinal_with cf card (n+1) by Def1;
  hence thesis by A2,ORDINAL2:50;
end;
