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

theorem  Th5:
  for f be c=-monotone Function-yielding Sequence st
     o in dom (f.B) & B in A
  holds
    o in dom union rng (f|A) &
    (union rng (f|A)).o = (union rng f).o
proof
  let f be c=-monotone Function-yielding Sequence;
  assume A1:o in dom (f.B) & B in A;
  A2: f.B<>{} by A1;
  then B in dom f by FUNCT_1:def 2;
  then A3: (f.B).o = (union rng f).o by A1,Th2;
  A4:(f|A).B = f.B by A1,FUNCT_1:49;
  then A5: B in dom (f|A) by A2,FUNCT_1:def 2;
  then dom ((f|A).B) c= dom union rng (f|A) by Th2;
  hence thesis by A3,A4,A5,Th2,A1;
end;
