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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/>



<a NAME="E1:66"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K5">rng</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_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>set </b><font color="Maroon">x</font> = <font color="Maroon">b</font>;<br/><a NAME="E1:66_1"/><i><font color="Green">E44</font></i>: 
<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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><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> <a href="relset_1.html#K9">|</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>
 <b>by </b><i><a class="ref" href="relat_1.html#T125">RELAT_1:125</a></i>;<br/><a NAME="E2:66_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> implies <span class="p1"><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="subset_1.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T17">ZFMISC_1:17</a></i>;<br/><a NAME="E3:66_1"/><b>then </b>
( <span class="p1"><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="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> <a href="hidden.html#R1">=</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="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</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="relset_1.html#K9">|</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> &amp; ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> implies <span class="p1"><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="subset_1.html#K9">/\</a> <span class="p1"><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="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i>, <a class="ref" href="relat_1.html#T127">RELAT_1:127</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:66_1"/><b>then </b>
(  <a href="relset_1.html#K4">dom</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> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &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; <span class="p1"><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="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> <a href="hidden.html#R1">=</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> &amp; ( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> implies ( <span class="p1"><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="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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &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 ) ) )
 <b>by </b><i><a class="ref" href="monoid_1.html#T1" target="_self">Th1</a>, , <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a>, <a class="ref" href="relat_1.html#T125">RELAT_1:125</a>, <a class="ref" href="relat_1.html#T137">RELAT_1:137</a></i>;<br/><b>hence </b><a NAME="E5:66_1"/>
<span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2"><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><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"><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>
 <b>by </b><i>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:66"/>
<span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2"><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><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


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