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<b>let </b><font color="Maroon" title="c1">D</font> be  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> ;<br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:81"/><i><font color="Green" title="E73">A1</font></i>: 
<font color="Maroon" title="c2">p1</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="c4">f</font>
 <b>and </b><br/><a NAME="E2:81"/><i><font color="Green" title="E74">A2</font></i>: 
<font color="Maroon" title="c3">p2</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"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">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" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> or <font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">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" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:81"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="finseq_6.html#T79" target="_self" title="FINSEQ_6:th.79">Th79</a></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" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><a NAME="E2:81_1_2"/><b>then </b>
not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:81_1_2"/><b>then </b><i><font color="Green" title="E75">A3</font></i>: 
not <font color="Maroon" title="c3">p2</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"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/><a NAME="E4:81_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K6" title="FINSEQ_4:func.6">-|</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:81"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="finseq_6.html#T44" target="_self" title="FINSEQ_6:th.44">Th44</a></i>;<br/><a NAME="E5:81_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K6" title="FINSEQ_4:func.6">-|</a> <font color="Maroon" title="c2">p1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/><a NAME="E6:81_1_2"/><b>then </b><i><font color="Green" title="E76">A4</font></i>: 
<font color="Maroon" title="c3">p2</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"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K6" title="FINSEQ_4:func.6">-|</a> <font color="Maroon" title="c2">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E2:81"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="finseq_6.html#E3:81_1_2"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:81_1_2"/><i><font color="Green" title="E77">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K6" title="FINSEQ_4:func.6">-|</a> <font color="Maroon" title="c2">p1</font></span>)</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> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:81"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="finseq_4.html#T51" title="FINSEQ_4:th.51">FINSEQ_4:51</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="finseq_4.html#K6" title="FINSEQ_4:func.6">-|</a> <font color="Maroon" title="c3">p2</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="finseq_6.html#E2:81"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="finseq_6.html#T44" target="_self" title="FINSEQ_6:th.44">Th44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K6" title="FINSEQ_4:func.6">-|</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K6" title="FINSEQ_4:func.6">-|</a> <font color="Maroon" title="c3">p2</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="finseq_6.html#E1:81"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="finseq_6.html#T44" target="_self" title="FINSEQ_6:th.44">Th44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K6" title="FINSEQ_4:func.6">-|</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="finseq_4.html#K6" title="FINSEQ_4:func.6">-|</a> <font color="Maroon" title="c3">p2</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="finseq_6.html#E6:81_1_2"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="finseq_6.html#T14" target="_self" title="FINSEQ_6:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K6" title="FINSEQ_4:func.6">-|</a> <font color="Maroon" title="c3">p2</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="finseq_6.html#E1:81"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="finseq_6.html#E6:81_1_2"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="finseq_6.html#T39" target="_self" title="FINSEQ_6:th.39">Th39</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p2</font>
<b>by </b><i><a class="txt" href="finseq_6.html#E6:81_1_2"><i><font color="Green" title="E76">A4</font></i></a>, <a class="txt" href="finseq_6.html#E7:81_1_2"><i><font color="Green" title="E77">A5</font></i></a>, <a class="ref" href="finseq_6.html#T44" target="_self" title="FINSEQ_6:th.44">Th44</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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