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

theorem Th81:
  for r be non-zero Sequence of REAL
    for p,s be Sequence
       for alpha be Ordinal st alpha c= dom r
         for x,y,z be Surreal st
    x<=y<=z & x in_meets_terms s,p,r,alpha & z in_meets_terms s,p,r,alpha
  holds y in_meets_terms s,p,r,alpha
proof
  let r be non-zero Sequence of REAL;
  let p,s be Sequence;
  let alpha be Ordinal such that alpha c= dom r;
  let x,y,z be Surreal such that
A1:x<=y<=z & x in_meets_terms s,p,r,alpha &
  z in_meets_terms s,p,r,alpha;
  let beta be Ordinal,sb,yb be Surreal such that
A2: beta in alpha & sb=s.beta & yb = p.beta;
  x is (sb,yb,r.beta)_term & z is (sb,yb,r.beta)_term by A1,A2;
  hence thesis by A1,Th74;
end;
