reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem Th101:
  for r be non-zero Sequence of REAL,
      y be strictly_decreasing uSurreal-Sequence st dom r=dom y holds
   r,y,dom r name_like Sum(r,y)
proof
  let r be non-zero Sequence of REAL,
      y be strictly_decreasing uSurreal-Sequence such that
A1: dom r=dom y;
  thus dom r c= dom r = dom y by A1;
  set P=Partial_Sums(r,y),s=Sum(r,y);
  let beta be Ordinal such that
A2:beta in dom r;
  let Pb be Surreal such that
A3: Pb = P.beta;
A4: dom P = succ (dom r/\dom r) by A1,Def17;
  y.beta in rng y by A1,A2,FUNCT_1:def 3;
  then reconsider yb = y.beta as uSurreal by SURREALO:def 12;
  P,y,r simplest_on_position dom r by A4,Def17,ORDINAL1:6;
  then s in_meets_terms P,y,r,dom r by A1;
  then
A5: s is (Pb,yb,r.beta)_term by A2,A3;
  then
A6: not s +- Pb ==0_No & omega-y (s +- Pb) == yb
  & omega-r (s + - Pb) = r.beta;
  not s == Pb
  proof
    assume s == Pb;
    then s +- Pb == Pb - Pb == 0_No by SURREALR:43,39;
    hence thesis by A6,SURREALO:4;
  end;
  hence thesis by A5,SURREALO:50;
end;
