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

theorem Th6:
  for f be c=-monotone Function-yielding Sequence
    for A,B be Ordinal st o in dom (f.B) & B in A
        holds
     (union rng (f|A)).o = (union rng f).o
proof
  let f be c=-monotone Function-yielding Sequence;
  let  A,B be Ordinal such that
  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;
  (f|A).B = f.B by A1,FUNCT_1:49;
  then  B in dom (f|A) by A2,FUNCT_1:def 2;
  hence thesis by A4,A3,Th2,A1;
end;
