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<b>given </b><font color="Maroon" title="c1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E1:62_1"/><i><font color="Green" title="E58">A3</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">a</font>] )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c2">X</font> =  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon">F<sub>1</sub></font>() as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="rewrite1.html#E1:62_1"><i><font color="Green" title="E58">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c1">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="rewrite1.html#E1:62_1"><i><font color="Green" title="E58">A3</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">Y</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 <font color="Maroon" title="c2">X</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> ;<br/>
<a NAME="E4:62_1"/><i><font color="Green" title="E59">A4</font></i>: 
<font color="Maroon" title="c3">a</font> <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 <font color="Maroon" title="c2">X</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="rewrite1.html#E1:62_1"><i><font color="Green" title="E58">A3</font></i></a></i>;<br/>
<a NAME="E5:62_1"/>
<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 <font color="Maroon" title="c2">X</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:62_1_1"/>
<font color="Maroon" title="c5">y</font> <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 <font color="Maroon" title="c2">X</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 ;<br/>

<a NAME="E2:62_1_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 <font color="Maroon" title="c2">X</font> st <br/>( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 
;<br/>
<b>hence </b><a NAME="E3:62_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:62_1"/><i><font color="Green" title="E60">A5</font></i>: 
<font color="Maroon" title="c5">a</font> <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 <font color="Maroon" title="c2">X</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 <b>and </b><br/><a NAME="E8:62_1"/><i><font color="Green" title="E61">A6</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>] </span>}</span>  &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/> not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">b</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">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:62"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="rewrite1.html#E4:62_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="rewrite1.html#D16" target="_self" title="REWRITE1:def.16">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E10:62_1"/><i><font color="Green" title="E62">A7</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>] )
 <b>by </b><i><a class="txt" href="rewrite1.html#E7:62_1"><i><font color="Green" title="E60">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:62_1_2"/><i><font color="Green" title="E63">A8</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</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">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">b</font>] )
 ;<br/><a NAME="E2:62_1_2"/><b>then </b>
( not <font color="Maroon" title="c7">b</font> <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 <font color="Maroon" title="c2">X</font> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E8:62_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><b>hence </b><a NAME="E3:62_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:62_1_2"><i><font color="Green" title="E63">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:62_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rewrite1.html#E2:62"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="rewrite1.html#E10:62_1"><i><font color="Green" title="E62">A7</font></i></a></i>;<br/>


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