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 Th42:
  x in L_(sReal.r) iff ex n st x = uDyadic.([/ r*(2|^n)-1\] / (2|^n))
proof
  sReal.r = [(the set of all uDyadic.( [/ r*(2|^k)-1 \] / (2|^k))
  where k is Nat),the set of all uDyadic.( [\ r*(2|^m)+1 /] / (2|^m) )]
  by Def6;
  hence thesis;
end;
