<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:23_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E2:23_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E4:23_1_1"/>
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:23_1_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
;<br/>
<a NAME="E6:23_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E7:23_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:23_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:23_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r2</font> where <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span> 
 
;<br/>
<a NAME="E3:23_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
 ex <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font> &amp; <font color="Olive">g2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 
;<br/>
<a NAME="E4:23_1_2"/><b>then </b>
for <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a>  holds <br/>(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font> or  not <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g2</font> )
 
;<br/>
<a NAME="E5:23_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g1</font> </span>}</span> 
 
;<br/>
<a NAME="E6:23_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font>
 
;<br/>
<b>hence </b><a NAME="E7:23_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:23"/>
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:23_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E2:23_2_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E4:23_2_1"/>
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:23_2_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
;<br/>
<a NAME="E6:23_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E7:23_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D15">PROB_1:def 15</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:23_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:23_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r2</font> where <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="prob_1.html#D15">PROB_1:def 15</a></i>;<br/>
<a NAME="E3:23_2_2"/><b>then </b><i><font color="Green">E48</font></i>: 
 ex <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font> &amp; <font color="Olive">g2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 
;<br/>
<a NAME="E4:23_2_2"/><b>then </b>
for <font color="Olive">g1</font> being  <a href="real_1.html#NM1">Real</a>  holds <br/>(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g1</font> or  not <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g1</font> )
 
;<br/>
<a NAME="E5:23_2_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g</font> </span>}</span> 
 
;<br/>
<a NAME="E6:23_2_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font>
 
;<br/>
<b>hence </b><a NAME="E7:23_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:23"/>
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:23_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_3_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E2:23_3_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E4:23_3_1"/>
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="prob_1.html#D15">PROB_1:def 15</a></i>;<br/>
<a NAME="E5:23_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font>
 
;<br/>
<a NAME="E6:23_3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g1</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E7:23_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:23_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:23_3_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r2</font> where <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r2</font> </span>}</span> 
 
;<br/>
<a NAME="E3:23_3_2"/><b>then </b><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g2</font> )
 
;<br/>
<a NAME="E4:23_3_2"/><b>then </b>
for <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a>  holds <br/>(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font> or  not <font color="Olive">g2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 
;<br/>
<a NAME="E5:23_3_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
;<br/>
<a NAME="E6:23_3_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D15">PROB_1:def 15</a></i>;<br/>
<b>hence </b><a NAME="E7:23_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:23"/>
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3" target="_self">right_closed_halfline</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:23_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_4_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E2:23_4_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E4:23_4_1"/>
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:23_4_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x'</font>
 
;<br/>
<a NAME="E6:23_4_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g1</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E7:23_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:23_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:23_4_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r2</font> where <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r2</font> </span>}</span> 
 
;<br/>
<a NAME="E3:23_4_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 ex <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g2</font> )
 
;<br/>
<a NAME="E4:23_4_2"/><b>then </b>
for <font color="Olive">g1</font> being  <a href="real_1.html#NM1">Real</a>  holds <br/>(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g1</font> or  not <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 
;<br/>
<a NAME="E5:23_4_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span> 
 
;<br/>
<a NAME="E6:23_4_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font>
 
;<br/>
<b>hence </b><a NAME="E7:23_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:23"/>
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4" target="_self">right_open_halfline</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
