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<b>thus </b><a NAME="E1:11"/>
 <a href="numbers.html#K1" target="_self">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K2" target="_self">COMPLEX</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a><br/>

<a NAME="E2:11"/><i><font color="Green">E13</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 0,1</span>)</span> <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>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<a NAME="E3:11"/><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 0,1</span>)</span> <a href="tarski.html#R1">c=</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="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<a NAME="E4:11"/><i><font color="Green">E28</font></i>: 
 <a href="arytm_2.html#K2">REAL+</a>  <a href="tarski.html#R1">c=</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K13">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:11"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="arytm_3.html#K13">{}</a> ,<a href="arytm_3.html#K13">{}</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#T3">ARYTM_2:3</a></i>;<br/>
<a NAME="E6:11"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="arytm_2.html#K2">REAL+</a>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1" target="_self">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T40">ZFMISC_1:40</a></i>;<br/>
<a NAME="E7:11"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1" target="_self">REAL</a> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#T21">ARYTM_2:21</a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E8:11"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 0,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1" target="_self">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:11"/><b>then </b><i><font color="Green">E30</font></i>: 
0,1 <a href="funct_4.html#K5">--&gt;</a> 0,1 <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,<a href="arytm_3.html#K14">one</a> </span><a href="tarski.html#K2">}</a></span>,<a href="numbers.html#K1" target="_self">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = 0, <font color="Maroon">j</font> = 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1" target="_self">REAL</a>  <b>by </b><i><a class="txt" href="numbers.html#E1"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="arytm_2.html#T21">ARYTM_2:21</a></i>;<br/>
<a NAME="E11:11"/><i><font color="Green">E31</font></i>: 
not 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">z</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1" target="_self">REAL</a> 
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">X</font> = <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,<a href="arytm_3.html#K14">one</a> </span><a href="tarski.html#K2">}</a></span>,<a href="numbers.html#K1" target="_self">REAL</a>  : <font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <a href="arytm_3.html#K14">one</a>  <a href="hidden.html#R1">=</a> 0 </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1"/>
0,1 <a href="funct_4.html#K5">--&gt;</a> 0,1 <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,<a href="arytm_3.html#K14">one</a> </span><a href="tarski.html#K2">}</a></span>,<a href="numbers.html#K1" target="_self">REAL</a>  : <font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <a href="arytm_3.html#K14">one</a>  <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 ;<br/><a NAME="E2:11_1"/><b>then </b>
 ex <font color="Olive">x</font> being   <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,<a href="arytm_3.html#K14">one</a> </span><a href="tarski.html#K2">}</a></span>,<a href="numbers.html#K1" target="_self">REAL</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> 0,1 &amp; <font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <a href="arytm_3.html#K14">one</a>  <a href="hidden.html#R1">=</a> 0 )
 ;<br/><b>hence </b><a NAME="E3:11_1"/>
contradiction
 <b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:11"/><b>then </b>
0,1 <a href="funct_4.html#K5">--&gt;</a> 0,1 <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,<a href="arytm_3.html#K14">one</a> </span><a href="tarski.html#K2">}</a></span>,<a href="numbers.html#K1" target="_self">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,<a href="arytm_3.html#K14">one</a> </span><a href="tarski.html#K2">}</a></span>,<a href="numbers.html#K1" target="_self">REAL</a>  : <font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <a href="arytm_3.html#K14">one</a>  <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 
<b>by </b><i><a class="txt" href="numbers.html#E3"><i><font color="Green">Lemma12</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:11"/>
 <a href="numbers.html#K1" target="_self">REAL</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="numbers.html#K2" target="_self">COMPLEX</a> 
 <b>by </b><i><a class="txt" href="numbers.html#E2:11"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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