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<b>let </b><font color="Maroon">F</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">i2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">xi1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Ai2</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E20</font></i>: 
(  <a href="card_3.html#K4">product</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">xi1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> )
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E3:8"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:8"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">Ai2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 ;<br/>

<a NAME="E5:8"/>
not <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ai2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">xi2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:8"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">xi2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>and </b><br/><a NAME="E8:8"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">xi2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xi2</font> = <font color="Maroon">xi2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">f'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">F</font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E12:8"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E13:8"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E14:8"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E15:8"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i2</font>,<font color="Maroon">xi2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi2</font>
 
<b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<b>thus </b><a NAME="E16:8"/>
( <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i1</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">xi1</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i2</font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">Ai2</font> implies ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">xi1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i1</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">xi1</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i2</font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">Ai2</font>
 ;<br/>

<b>thus </b><a NAME="E2:8_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font>
 ;<br/>

<a NAME="E2:8_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i2</font>,<font color="Maroon">xi2</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="yellow17.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E3:8_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i2</font>,<font color="Maroon">xi2</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i1</font>,<font color="Maroon">xi1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i1</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">xi1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:8_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i2</font>,<font color="Maroon">xi2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i2</font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">Ai2</font>
 
<b>by </b><i>, , , <a class="txt" href="yellow17.html#E1:8"><i><font color="Green">E20</font></i></a>, , </i>;<br/>
<a NAME="E5:8_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i2</font>,<font color="Maroon">xi2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i2</font>,<font color="Maroon">xi2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai2</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E6:8_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E3:8_1"/>
<font color="Maroon">xi1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i1</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:8_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">xi1</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E5:8_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">xi1</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ai2</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T158">FUNCT_1:158</a></i>;<br/>
<b>hence </b><a NAME="E6:8_1"/>
<font color="Maroon">xi1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E17:8"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>and </b><br/><a NAME="E18:8"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">xi1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai2</font>
 ;<br/>

<a NAME="E19:8"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">xi1</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ai2</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E20:8"/>
<span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i1</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">xi1</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <font color="Maroon">F</font>,<font color="Maroon">i2</font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">Ai2</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>


</div>
