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

theorem Th7:
   x,y are_commensurate iff
     ex n be positive Nat st x < uInt.n * y & y < uInt.n * x
proof
  thus x,y are_commensurate implies
     ex n be positive Nat st x < uInt.n * y & y < uInt.n * x
  proof
    assume
A1: x,y are_commensurate;
A2: x is positive & y is positive by A1,Th3;
    consider n be positive Nat such that
A3: x < uInt.n * y by A1;
    consider m be positive Nat such that
A4: y < uInt.m * x by A1;
    take n+m;
    n+0 < n+m & m+0 < n+m by XREAL_1:6;
    then uInt.n * y <= uInt.(n+m)* y & uInt.m * x <= uInt.(n+m)* x
    by A2,SURREALR:70,SURREALN:9;
    hence thesis by A3,A4,SURREALO:4;
  end;
  thus thesis;
end;
