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
    for y be strictly_decreasing Surreal-Sequence
      for alpha be Ordinal st r,y,alpha name_like x
  holds r|alpha,y|alpha,alpha name_like x
proof
  let r be non-zero Sequence of REAL;
  let y be strictly_decreasing Surreal-Sequence;
  let A be Ordinal such that
A1: r,y,A name_like x;
A2: dom (r|A) = A & dom (y|A) = A by A1,RELAT_1:62;
  thus A c= dom (r|A) = dom (y|A) by A2;
  let B be Ordinal such that
A3: B in A;
  let Pb be Surreal such that
A4: Pb = Partial_Sums(r|A,y|A).B;
A5: Partial_Sums(r|A,y|A) = Partial_Sums(r,y)|succ A by Th85;
  B in succ A by A3,ORDINAL1:8;
  then Pb = Partial_Sums(r,y).B by A4,A5,FUNCT_1:49;
  then not x == Pb & r.B = omega-r (x - Pb) &
  y.B = omega-y (x - Pb) by A1,A3;
  hence thesis by A3,FUNCT_1:49;
end;
