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

theorem Th74:
  for s,p be Surreal st r <>0
      for x,y,z be Surreal st x is (s,p,r)_term & z is (s,p,r)_term &
          x <= y <= z
  holds y is (s,p,r)_term
proof
  let s,p be Surreal such that
A1: r <>0;
  let x,y,z be Surreal such that
A2:x is (s,p,r)_term & z is (s,p,r)_term and
A3:x <= y <= z;
  set N =No_omega^ p, R = uReal.r,sNR=s+R*N;
  set X = x - sNR,Y = y -sNR, Z=z-sNR;
  |.X.| infinitely< N & |.Z.| infinitely< N by A2,Th73;
  then
A4: |.X.|+|.Z.| infinitely< N by Th18;
  X <= Y <= Z by A3,SURREALR:43;
  then |.Y.| infinitely< N by A4,Th11,Th51;
  hence thesis by A1,Th73;
end;
