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<div class="add">

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

<b>set </b><font color="Maroon">X</font> =  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">a</font>;<br/>
<a NAME="E1:32"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">a</font> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T17">BORSUK_5:17</a></i>;<br/>
<a NAME="E2:32"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:32_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T17">XREAL_1:17</a></i>;<br/>
<a NAME="E3:32_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
;<br/>

<b>thus </b><a NAME="E4:32_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 )
 <b>by </b><i>, <a class="ref" href="borsuk_5.html#T17">BORSUK_5:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


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