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 Th59:
  for n be positive Nat holds
    x - (uInt.n)" < real_qua x < x+ (uInt.n)"
proof
  let n be positive Nat;
    x- (uInt.n)" in
       the set of all x - (uInt.k)" where k is positive Nat;
  then
A1: x+- (uInt.n)" in L_real_qua x by Def8;
  x+ (uInt.n)" in the set of all x + (uInt.k)" where k is positive Nat;
  then
A2: x+ (uInt.n)" in R_real_qua x by Def8;
  L_real_qua x << {real_qua x} << R_real_qua x & real_qua x in {real_qua x}
  by TARSKI:def 1,SURREALO:11;
  hence thesis by A1,A2;
end;
