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

theorem Th103:
  for r be non-zero Sequence of REAL,
      y be strictly_decreasing uSurreal-Sequence,
      A be Ordinal st A c= dom r = dom y
    for x,z be Surreal st r,y,A name_like x & x == z holds
      r,y,A name_like z
proof
  let r be non-zero Sequence of REAL,
      y be strictly_decreasing uSurreal-Sequence,
      A be Ordinal such that A c= dom r=dom y;
  let x,z be Surreal such that A2: r,y,A name_like x & x == z;
  thus A c= dom r = dom y by A2;
  let B be Ordinal such that A3: B in A;
  let Pb be Surreal such that A4:Pb = Partial_Sums(r,y).B;
  thus not z == Pb by A2,A3,A4,SURREALO:10;
A5: x - Pb == z - Pb by A2,SURREALR:66;
A6: not x == Pb & r.B = omega-r (x - Pb) & y.B = omega-y (x - Pb) by A2,A3,A4;
  then not x -Pb == 0_No by SURREALR:45,46;
  hence thesis by A5,A6,Th70;
end;
