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<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:55_1_1"/><i><font color="Green" title="E14">W1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E3:55_1_1"/><i><font color="Green" title="E15">W2</font></i>: 
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> <font color="Maroon" title="c1">R</font> holds <br/><font color="Maroon" title="c2">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#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:55_1_1_1"/>
( <font color="Maroon" title="c3">y</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> <font color="Maroon" title="c2">f</font> implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_1_1_1"/>
<font color="Maroon" title="c3">y</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> <font color="Maroon" title="c2">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:55_1_1_1_1"/><i><font color="Green" title="E16">W3</font></i>: 
<font color="Maroon" title="c4">a</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="c2">f</font>
 <b>and </b><br/><a NAME="E4:55_1_1_1_1"/><i><font color="Green" title="E17">W4</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
;<br/>

<a NAME="E5:55_1_1_1_1"/><i><font color="Green" title="E18">F</font></i>: 
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="finset_1.html#E2:55_1_1"><i><font color="Green" title="E14">W1</font></i></a>, <a class="txt" href="finset_1.html#E3:55_1_1_1_1"><i><font color="Green" title="E16">W3</font></i></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E6:55_1_1_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<b>by </b><i><a class="txt" href="finset_1.html#E3:55_1_1"><i><font color="Green" title="E15">W2</font></i></a>, <a class="txt" href="finset_1.html#E3:55_1_1_1_1"><i><font color="Green" title="E16">W3</font></i></a>, <a class="txt" href="finset_1.html#E4:55_1_1_1_1"><i><font color="Green" title="E17">W4</font></i></a>, <a class="txt" href="finset_1.html#E2:55_1_1"><i><font color="Green" title="E14">W1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:55_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E2:55_1_1"><i><font color="Green" title="E14">W1</font></i></a>, <a class="txt" href="finset_1.html#E3:55_1_1_1_1"><i><font color="Green" title="E16">W3</font></i></a>, <a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a>, <a class="txt" href="finset_1.html#E5:55_1_1_1_1"><i><font color="Green" title="E18">F</font></i></a></i>;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:55_1_1_1"/><i><font color="Green" title="E16">G</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 ;<br/><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="finset_1.html#E3:55_1_1"><i><font color="Green" title="E15">W2</font></i></a>, <a class="txt" href="finset_1.html#E3:55_1_1_1"><i><font color="Green" title="E16">G</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E5:55_1_1_1"/>
<font color="Maroon" title="c3">y</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> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E2:55_1_1"><i><font color="Green" title="E14">W1</font></i></a>, <a class="txt" href="finset_1.html#E3:55_1_1_1"><i><font color="Green" title="E16">G</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:55_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:55_1_1"/>
 <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c1">R</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="finset_1.html#E2:55_1_1"><i><font color="Green" title="E14">W1</font></i></a>, <a class="ref" href="finset_1.html#T26" target="_self" title="FINSET_1:th.26">Th26</a></i>;<br/>


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