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<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_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"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  or  not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>suppose </b></a><a NAME="E1:14_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">c</font>, <font color="Maroon">z</font> = 0 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><b>take </b>
<font color="Maroon">r</font>
;<br/><b>take </b>
<font color="Maroon">z</font>
;<br/><b>thus </b><a NAME="E3:14_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5" target="_self">[*</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">z</font></span><a href="arytm_0.html#K5" target="_self">*]</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><b>suppose </b></a><a NAME="E1:14_1_2"/>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><div class="add"><a NAME="E2:14_1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>,<a href="numbers.html#K1">REAL</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>,<a href="numbers.html#K1">REAL</a>  : <font color="Olive">x</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 <b>by </b><i><a class="ref" href="numbers.html#D2">NUMBERS:def 2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:14_1_2"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>,<a href="numbers.html#K1">REAL</a> 
 ;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:14_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E6:14_1_2"/><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 <b>and </b><br/><a NAME="E7:14_1_2"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E8:14_1_2"/>
( 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E9:14_1_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="arytm_0.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0, <font color="Maroon">s</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><a NAME="E11:14_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T4" target="_self">Th4</a>, <a class="ref" href="arytm_0.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_4.html#T69">FUNCT_4:69</a></i>;<br/><b>take </b>
<font color="Maroon">r</font>
;<br/><b>take </b>
<font color="Maroon">s</font>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:14_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><a NAME="E3:14_1_2_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>,<a href="numbers.html#K1">REAL</a>  : <font color="Olive">x</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:14_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:14_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5" target="_self">[*</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="arytm_0.html#K5" target="_self">*]</a></span>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E3:14_1_2"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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