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

theorem
  for x,y,z be Surreal st
    x,z are_commensurate & x <= y <= z holds
  x,y are_commensurate & y,z are_commensurate
proof
  let x,y,z be Surreal such that
A1:x,z are_commensurate & x <= y <= z;
A2:x is positive & z is positive by A1,Th3;
  per cases by A1;
  suppose x==y or y==z;
    hence thesis by A1,Th5,A2,Th2;
  end;
  suppose
A3: x < y < z;
    consider n be positive Nat such that
A4: z < uInt.n * x by A1;
    y <= z by A3;
    then
A5: y < uInt.n * x by A4,SURREALO:4;
    0_No < uInt.n by SURREALI:def 8;
    then uInt.n * x <= uInt.n * y by A3,SURREALR:70;
    then
A6: z < uInt.n * y by A4,SURREALO:4;
A7: y < uInt.1*z by A3,SURREALN:11;
    y = uInt.1*y by SURREALN:11;
    hence thesis by A3,A7,A6,A5;
  end;
end;
