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

theorem Th2:
  for f be c=-monotone Function-yielding Sequence
    for A st A in dom f holds
      dom (f.A) c= dom union rng f &
  for o st o in dom (f.A) holds (f.A).o = (union rng f).o
proof
  let f be c=-monotone Function-yielding Sequence;
  set U=union rng f;
  let A be Ordinal such that A1: A in dom f;
  A2: f.A in rng f by A1,FUNCT_1:def 3;
  thus dom (f.A) c= dom U
  proof
    let y be object;
    assume y in dom (f.A);
    then [y,f.A.y] in f.A by FUNCT_1:def 2;
    then [y,f.A.y] in U by A2,TARSKI:def 4;
    hence thesis by FUNCT_1:1;
  end;
  let y be object;
  assume y in dom (f.A);
  then [y,f.A.y] in f.A by FUNCT_1:def 2;
  then [y,f.A.y] in U by A2,TARSKI:def 4;
  hence thesis by FUNCT_1:1;
end;
