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





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><b>reconsider </b><font color="Maroon">pa</font> = <font color="Maroon">p</font> <a href="monoid_0.html#K1">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font> ;<br/><a NAME="E2:67_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:67_1"/><b>then </b>
( not <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="relset_1.html#K9">|</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="funct_1.html#T89">FUNCT_1:89</a></i>;<br/><a NAME="E4:67_1"/><b>then </b>
( not <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="relset_1.html#K9">|</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="relset_1.html#K9">|</a> <font color="Maroon">p</font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 ;<br/><a NAME="E5:67_1"/><b>then </b>
( <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="monoid_0.html#K1">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="relset_1.html#K9">|</a> <font color="Maroon">pa</font></span>)</span></span>)</span> &amp; <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">p</font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="relset_1.html#K9">|</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="relset_1.html#K9">|</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="monoid_0.html#K1">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="relset_1.html#K9">|</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">a</font></span>)</span> <a href="monoid_1.html#K15" target="_self">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 &amp; ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> implies (  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="relset_1.html#K9">|</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="monoid_0.html#K1">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="relset_1.html#K9">|</a> <font color="Maroon">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">a</font></span>)</span> <a href="monoid_1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) ) &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="relset_1.html#K9">|</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="relset_1.html#K9">|</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> ) &amp; <span class="p1">(<span class="default"><span class="p2"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">p</font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p2">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="monoid_1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">p</font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_1.html#K15" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">a</font></span>)</span> <a href="monoid_1.html#K15" target="_self">.</a> <font color="Maroon">b</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/><b>hence </b><a NAME="E6:67_1"/>
<span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="monoid_0.html#K1">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">p</font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p2">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="monoid_1.html#K15" target="_self">.</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:67"/>
<span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="monoid_0.html#K1">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">p</font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i/>;<br/>


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