reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;
reserve d,d1,d2 for Dyadic;
reserve i,j for Integer,
        n,m,p for Nat;
reserve r,r1,r2 for Real;

theorem Th41: :::
  [/ r*(2|^n)-1\] / (2|^n) < r < [\ r*(2|^n)+1/] / (2|^n)
proof
  [/ r*(2|^n)-1\] < r*(2|^n)-1+1 by INT_1:def 7;
  then
A1:[/ r*(2|^n)-1\]/(2|^n) < (r*(2|^n))/(2|^n) by XREAL_1:74;
  r*(2|^n)+1 -1 < [\ r*(2|^n)+1/] by INT_1:def 6;
  then (r*(2|^n))/(2|^n) < [\ r*(2|^n)+1/] /(2|^n)
  by XREAL_1:74;
  hence thesis by A1,XCMPLX_1:89;
end;
