reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;

theorem Th46:
  REAL is non bounded_below non bounded_above
proof
  thus REAL is non bounded_below
  proof
    given r being Real such that
A1: r is LowerBound of REAL;
    consider s being Real such that
A2: s < r by XREAL_1:2;
    s in REAL by XREAL_0:def 1;
    hence contradiction by A1,A2,Def2;
  end;
  given r being Real such that
A3: r is UpperBound of REAL;
  consider s being Real such that
A4: r < s by XREAL_1:1;
  s in REAL by XREAL_0:def 1;
  hence contradiction by A3,A4,Def1;
end;
