<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c2">i1</font>, <font color="Maroon" title="c3">i2</font>, <font color="Maroon" title="c4">xi1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c5">Ai2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green" title="E9">A1</font></i>: 
(  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i1</font> )
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c2">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font>
 <b>and </b><br/><a NAME="E3:8"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c3">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font>
 <b>and </b><br/><a NAME="E4:8"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c5">Ai2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font>
 ;<br/>

<a NAME="E5:8"/>
not <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Ai2</font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E4:8"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">xi2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:8"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c6">xi2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font>
 <b>and </b><br/><a NAME="E8:8"/><i><font color="Green" title="E14">A6</font></i>: 
not <font color="Maroon" title="c6">xi2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Ai2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">xi2</font> = <font color="Maroon" title="c6">xi2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font> <b>by </b><i><a class="txt" href="yellow17.html#E7:8"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">f'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font>;<br/>
<b>consider </b><font color="Maroon" title="c9">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:8"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c8">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 <b>and </b><br/><a NAME="E13:8"/><i><font color="Green" title="E16">A8</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font>
 <b>and </b><br/><a NAME="E14:8"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E15:8"/><i><font color="Green" title="E17">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c7">xi2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">xi2</font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E3:8"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="yellow17.html#E13:8"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">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" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">i1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">xi1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c3">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c5">Ai2</font> implies ( <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i2</font> &amp; <font color="Maroon" title="c4">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">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" title="E18">A10</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">i1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">xi1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c3">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c5">Ai2</font>
 ;<br/>

<b>thus </b><a NAME="E2:8_1"/><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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" title="E20">A12</font></i>: 
<font color="Maroon" title="c2">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i2</font>
 ;<br/>

<a NAME="E2:8_1_1"/><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c7">xi2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="yellow17.html#E7:8"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="yellow17.html#E12:8"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="yellow17.html#T2" target="_self" title="YELLOW17:th.2">Th2</a></i>;<br/>
<a NAME="E3:8_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c7">xi2</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c4">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">i1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">xi1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="yellow17.html#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="yellow17.html#E2:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="yellow17.html#E7:8"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="yellow17.html#E12:8"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="yellow17.html#T2" target="_self" title="YELLOW17:th.2">Th2</a>, <a class="ref" href="yellow17.html#T5" target="_self" title="YELLOW17:th.5">Th5</a></i>;<br/>
<a NAME="E4:8_1_1"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c7">xi2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c3">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c5">Ai2</font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="yellow17.html#E2:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="yellow17.html#E1:8_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="yellow17.html#E1:8_1_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="yellow17.html#E2:8_1_1"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="yellow17.html#T6" target="_self" title="YELLOW17:th.6">Th6</a></i>;<br/>
<a NAME="E5:8_1_1"/><b>then </b>
( <font color="Maroon" title="c9">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c7">xi2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c3">i2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c3">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c7">xi2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Ai2</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E6:8_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow17.html#E8:8"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="yellow17.html#E15:8"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="pralg_3.html#D2" title="PRALG_3:def.2">PRALG_3:def 2</a></i>;<br/>


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

<a NAME="E3:8_1"/>
<font color="Maroon" title="c4">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">i1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow17.html#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="yellow17.html#E2:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="yellow17.html#T3" target="_self" title="YELLOW17:th.3">Th3</a></i>;<br/>
<a NAME="E4:8_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">xi1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">i1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E5:8_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">xi1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Ai2</font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E1:8_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="yellow17.html#E2:8_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="funct_1.html#T158" title="FUNCT_1:th.158">FUNCT_1:158</a></i>;<br/>
<b>hence </b><a NAME="E6:8_1"/>
<font color="Maroon" title="c4">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Ai2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">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" title="E18">A14</font></i>: 
<font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i2</font>
 <b>and </b><br/><a NAME="E18:8"/><i><font color="Green" title="E19">A15</font></i>: 
<font color="Maroon" title="c4">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Ai2</font>
 ;<br/>

<a NAME="E19:8"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">xi1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Ai2</font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E18:8"><i><font color="Green" title="E19">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">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" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">i1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">xi1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c3">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c5">Ai2</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E17:8"><i><font color="Green" title="E18">A14</font></i></a>, <a class="ref" href="relat_1.html#T178" title="RELAT_1:th.178">RELAT_1:178</a></i>;<br/>


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