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 Th26:
  dom phi = M & rng phi c= N & M in cf N implies sup phi in N &
  Union phi in N
proof
  assume that
A1: dom phi = M and
A2: rng phi c= N and
A3: M in cf N;
  card rng phi c= card M by A1,CARD_1:12;
  then card rng phi in cf N by A3,ORDINAL1:12;
  hence thesis by A2,Th25;
end;
