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<b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfunct_1.html#K2" target="_self">^</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfunct_1.html#K2" target="_self">^</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E3:12"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfunct_1.html#K2" target="_self">^</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfunct_1.html#K2" target="_self">^</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E5:12"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfunct_1.html#K2" target="_self">^</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:12"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K12">"</a>  <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:12"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:12"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E9:12"/><b>then </b>
not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="partfun2.html#T44">PARTFUN2:44</a></i>;<br/>
<a NAME="E10:12"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:12"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/>


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