reserve x,y,z,w for ExtReal,
  r for Real;

theorem Th3:
  x < z implies ex y being Real st x < y & y < z
proof
  assume
A1: x < z;
  per cases by XXREAL_0:14;
  suppose x in REAL & z in REAL;
    hence thesis by A1,XREAL_1:5;
  end;
  suppose x = +infty or z = -infty;
    hence thesis by A1,XXREAL_0:4,6;
  end;
  suppose
A2: z = +infty;
    per cases by A1,A2,XXREAL_0:14;
    suppose x = -infty;
      hence thesis by A2;
    end;
    suppose x in REAL;
      then reconsider x1 = x as Real;
      take x1+1;
A3:   x1+1 in REAL by XREAL_0:def 1;
      x1+0 < x1+1 by XREAL_1:8;
      hence thesis by A2,A3,XXREAL_0:9;
    end;
  end;
  suppose
A4: x = -infty;
    per cases by A1,A4,XXREAL_0:14;
    suppose z = +infty;
      hence thesis by A4;
    end;
    suppose z in REAL;
      then reconsider z1 = z as Real;
      take z1-1;
A5:   z1-1 in REAL by XREAL_0:def 1;
      z1-1 < z1-0 by XREAL_1:10;
      hence thesis by A4,A5,XXREAL_0:12;
    end;
  end;
end;
