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<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/><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>set </b><font color="Maroon">X</font> = <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>;<br/>
<b>assume </b><a NAME="E1:14"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:14"/><b>then </b><i><font color="Green">E34</font></i>: 
not <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E3:14"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> holds <br/><font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font>, <font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E5:14"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/>
<a NAME="E6:14"/>
for <font color="Olive">b</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">b</font> holds <br/> ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> &amp; <font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:14_1"/><i><font color="Green">E46</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:14_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> holds <br/><font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1"/>
( <font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> or <font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1"/>
<font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><b>hence </b><a NAME="E3:14_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T17">XREAL_1:17</a></i>;<br/><a NAME="E3:14_1_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/><a NAME="E4:14_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:14_1_1_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/><a NAME="E6:14_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/><a NAME="E7:14_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><b>hence </b><a NAME="E8:14_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E7:14"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


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