reserve A,B,C for Ordinal,
        o for object,
        x,y,z,t,r,l for Surreal,
        X,Y for set;

theorem Th1:
  for f be c=-monotone Function-yielding Sequence
    for o be object st o in dom union rng f
      ex A st A in dom f & o in dom (f.A)
proof
  let f be c=-monotone Function-yielding Sequence;
  set U=union rng f;
  let x be object;
  assume x in dom U;
  then [x,U.x] in U by FUNCT_1:def 2;
  then consider Y be set such that
  A1:[x,U.x] in Y & Y in rng f by TARSKI:def 4;
  consider z be object such that
  A2:z in dom f & f.z = Y by FUNCT_1:def 3,A1;
  reconsider z as Ordinal by A2;
  take z;
  thus thesis by A2,A1,FUNCT_1:1;
end;
