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<b>let </b><font color="Maroon">D</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">p1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">p2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:81"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:81"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p1</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> or <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1"><b>suppose </b></a><a NAME="E1:81_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_2"><b>suppose </b></a><a NAME="E1:81_1_2"/>
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><a NAME="E2:81_1_2"/><b>then </b>
not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:81_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p1</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E4:81_1_2"/>
<font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p1</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:81_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><a NAME="E6:81_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_6.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:81_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T51">FINSEQ_4:51</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p2</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p2</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_6.html#T17" target="_self">Th17</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p2</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p2</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p1</font></span>)</span> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p2</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p2</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_6.html#T21" target="_self">Th21</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p2</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p2</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_6.html#T21" target="_self">Th21</a>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="finseq_6.html#T21" target="_self">Th21</a>, , </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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