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

theorem Th75:
  for r be Sequence of REAL
    for y,s1,s2 be Sequence
      for alpha be Ordinal st s1|alpha = s2|alpha &
         x in_meets_terms s1,y,r,alpha holds
  x in_meets_terms s2,y,r,alpha
proof
  let r be Sequence of REAL;
  let y,s1,s2 be Sequence;
  let alpha be Ordinal such that
A1: s1|alpha = s2|alpha and
A2: x in_meets_terms s1,y,r,alpha;
  let beta be Ordinal,sb,yb be Surreal such that
A3: beta in alpha & sb=s2.beta & yb = y.beta;
  s2.beta = (s2|alpha).beta & s1.beta = (s1|alpha).beta by A3,FUNCT_1:49;
  hence x is (sb,yb,r.beta)_term by A3,A2,A1;
end;
