 reserve A,B,O for Ordinal,
      n,m for Nat,
      a,b,o for object,
      x,y,z for Surreal,
      X,Y,Z for set,
      Inv,I1,I2 for Function;

theorem Th27:
  for f be Function st dom f = NAT & y in Union f
    ex n st y in f.n &
       for m st y in f.m holds n <= m
proof
  let f be Function such that
A1:dom f = NAT & y in Union f;
  defpred P[Nat] means y in f.$1;
  consider n be object such that
A2:n in dom f & y in f.n by A1,CARD_5:2;
  reconsider n as Nat by A1,A2;
  y in f.n by A2;
  then
A3:ex n st P[n];
  ex k be Nat st P[k] &
  for n being Nat st P[n] holds k <= n from NAT_1:sch 5(A3);
  hence thesis;
end;
