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

theorem
  for r be non-zero Sequence of REAL,
       y be strictly_decreasing Surreal-Sequence st
       x in_meets_terms Partial_Sums(r,y),y,r,dom r/\dom y &
       z in_meets_terms Partial_Sums(r,y),y,r,dom r/\dom y &
  not x == z
     for A be Ordinal,yA be Surreal st A in dom r/\dom y & yA = y.A holds
         omega-y (x - z) < yA
proof
  let r be non-zero Sequence of REAL;
  let y be strictly_decreasing Surreal-Sequence;
  set s = Partial_Sums(r,y),D=dom r/\dom y;
  assume that
A1: x in_meets_terms s,y,r,D &
  z in_meets_terms s,y,r,D and
A2: not x == z;
  let A be Ordinal,yA be Surreal;
  assume
A3: A in D & yA=y.A;
A4: not x - z == 0_No by SURREALR:47,A2;
  dom Partial_Sums(r,y) =succ D by Def17;
  then
A5: |.x -z.| infinitely< No_omega^ yA by XBOOLE_1:7,A3,A1,Th90;
  |.x -z.|, No_omega^ omega-y (x+-z) are_commensurate by A4,Def7;
  then No_omega^ omega-y (x +-z) < No_omega^ yA by Th9,A5,Th15;
  hence thesis by Lm5;
end;
