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

theorem
  0 < x implies ex y being Real st 0 < y & y + y < x
proof
  assume 0 < x;
  then consider x1 being Real such that
A1: 0 < x1 and
A2: x1 < x by Th3;
  consider x2 being Real such that
A3: 0 < x2 and
A4: x1 + x2 < x by A1,A2,Th49;
  take y = min(x1,x2);
  per cases;
  suppose
A5: x1 <= x2;
    hence 0 < y by A1,XXREAL_0:def 9;
    y = x1 by A5,XXREAL_0:def 9;
    then y + y <= x1 + x2 by A5,Th36;
    hence thesis by A4,XXREAL_0:2;
  end;
  suppose
A6: x2 <= x1;
    hence 0 < y by A3,XXREAL_0:def 9;
    y = x2 by A6,XXREAL_0:def 9;
    then y + y <= x1 + x2 by A6,Th36;
    hence thesis by A4,XXREAL_0:2;
  end;
end;
