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

<b>let </b><font color="Maroon">X</font> be  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>assume </b><a NAME="E2:17"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>suppose </b></a><a NAME="E1:17_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">g</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:17_1_1"/><i><font color="Green">E39</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T40">RCOMP_1:40</a></i>;<br/><a NAME="E4:17_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E5:17_1_1"/>
<font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T138">XREAL_1:138</a></i>;<br/><a NAME="E6:17_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:17_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E8:17_1_1"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:17_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> is   <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="E10:17_1_1"/>
<font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/><a NAME="E11:17_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> &amp; <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E12:17_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan5a.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E13:17_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><a NAME="E14:17_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i/>;<br/><a NAME="E15:17_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">l</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E17:17_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 ;<br/><b>thus </b><a NAME="E18:17_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><b>suppose </b></a><a NAME="E1:17_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">g</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:17_1_2"/><i><font color="Green">E55</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T40">RCOMP_1:40</a></i>;<br/><a NAME="E4:17_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E5:17_1_2"/>
<font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T138">XREAL_1:138</a></i>;<br/><a NAME="E6:17_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:17_1_2"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:17_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/><a NAME="E9:17_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E10:17_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> is   <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="E11:17_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E12:17_1_2"/>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan5a.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:17_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E14:17_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><a NAME="E15:17_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i/>;<br/><a NAME="E16:17_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">l</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E18:17_1_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 ;<br/><b>thus </b><a NAME="E19:17_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
