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

theorem Th104:
  for r be non-zero Sequence of REAL,
      y be strictly_decreasing uSurreal-Sequence,
      x be Surreal st dom r = dom y & Sum(r,y) == x
  holds name-ord x = dom r
proof
  let r be non-zero Sequence of REAL,
      y be strictly_decreasing uSurreal-Sequence,
      x be Surreal such that
A1: dom r = dom y & Sum(r,y) == x;
  consider r1 be non-zero Sequence of REAL,
           y1 be strictly_decreasing uSurreal-Sequence such that
A2: name-ord x = dom r1 = dom y1 & Sum(r1,y1) == x by Def21;
  Sum(r1,y1)==Sum(r,y) by A1,A2,SURREALO:10;
  hence thesis by A1,A2,Th102;
end;
