reserve a,b for R_eal;
reserve A,B for Interval;

theorem Th2:
  for a,b being R_eal st a < b ex x being R_eal st a < x & x < b & x in REAL
proof
  let a,b be R_eal;
A1: a in REAL or a in {-infty,+infty} by XBOOLE_0:def 3,XXREAL_0:def 4;
A2: b in REAL or b in {-infty,+infty} by XBOOLE_0:def 3,XXREAL_0:def 4;
  assume
A3: a < b;
  then
A4: ( not a = +infty)& not b = -infty by XXREAL_0:4,6;
  per cases by A1,A2,A4,TARSKI:def 2;
  suppose
    a in REAL & b in REAL;
    then consider x,y being Real such that
A5: x = a & y = b and
    x<=y by A3;
    consider z being Real such that
A6: x<z & z<y by A3,A5,XREAL_1:5;
    reconsider z as Element of REAL by XREAL_0:def 1;
    reconsider o = z as R_eal by XXREAL_0:def 1;
    take o;
    thus thesis by A5,A6;
  end;
  suppose
A7: a in REAL & b = +infty;
    then reconsider x = a as Real;
    consider z being Real such that
A8: x<z by XREAL_1:1;
    reconsider z as Element of REAL by XREAL_0:def 1;
    reconsider o = z as R_eal by XXREAL_0:def 1;
    take o;
    thus thesis by A7,A8,XXREAL_0:9;
  end;
  suppose
A9: a = -infty & b in REAL;
    then reconsider y = b as Real;
    consider z being Real such that
A10: z<y by XREAL_1:2;
    reconsider z as Element of REAL by XREAL_0:def 1;
    reconsider o = z as R_eal by XXREAL_0:def 1;
    take o;
    thus thesis by A9,A10,XXREAL_0:12;
  end;
  suppose
A11: a = -infty & b = +infty;
    take 0.;
    0. = In(0,REAL);
    hence thesis by A11;
  end;
end;
