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<b>thus </b><a NAME="E1:11"/>
 <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a><br/>

<a NAME="E2:11"/><i><font color="Green" title="E9">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>;<br/>
<a NAME="E3:11"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>;<br/>
<a NAME="E4:11"/><i><font color="Green" title="E11">A3</font></i>: 
 <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:11"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> ,<a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#T3" title="ARYTM_2:th.3">ARYTM_2:3</a></i>;<br/>
<a NAME="E6:11"/><b>then </b><i><font color="Green" title="E12">A4</font></i>: 
 <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="numbers.html#E4:11"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T40" title="ZFMISC_1:th.40">ZFMISC_1:40</a></i>;<br/>
<a NAME="E7:11"/><b>then </b>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#T21" title="ARYTM_2:th.21">ARYTM_2:21</a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E8:11"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="numbers.html#E3:11"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:11"/><b>then </b><i><font color="Green" title="E13">A5</font></i>: 
0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="numbers.html#E2:11"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c1">z</font> = 0, <font color="Maroon" title="c2">j</font> = 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="numbers.html#E6:11"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="arytm_2.html#T21" title="ARYTM_2:th.21">ARYTM_2:21</a></i>;<br/>
<a NAME="E11:11"/><i><font color="Green" title="E14">A6</font></i>: 
not 0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">z</font>,<font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="numbers.html#E11"><i><font color="Green" title="E7">Lm7</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="FUNCT_4:func.5">--&gt;</a> 0,1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span> 
 ;<br/><a NAME="E2:11_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,1 &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><b>hence </b><a NAME="E3:11_1"/>
contradiction
 <b>by </b><i><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">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" title="FUNCT_4:func.5">--&gt;</a> 0,1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span> 
 
<b>by </b><i><a class="txt" href="numbers.html#E9:11"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:11"/>
 <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 <b>by </b><i><a class="txt" href="numbers.html#E11:11"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


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