<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">A</font> be  <a href="unialg_1.html#V6">partial</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_1.html#L1">UAStr</a> ;<br/><b>let </b><font color="Maroon">P</font> be    <a href="pua2mss1.html#M3" target="_self">a_partition</a> of <font color="Maroon">A</font>;<br/>



<b>consider </b><font color="Maroon">EP</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:59"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EP</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T43">EQREL_1:43</a></i>;<br/>
<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a><br/>

<b>assume </b><a NAME="E3:59"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font> ;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">aa</font>;<br/>
<b>take </b>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E6:59"/>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>

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

<b>assume </b><a NAME="E7:59"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">aa</font> ;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">EP</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E10:59"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a><br/>

<b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EP</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> =  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EP</font>,<font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font> <b>by </b><i><a class="txt" href="pua2mss1.html#E10:59"><i><font color="Green">E36</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E4:59_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<a NAME="E5:59_1"/>
for <font color="Olive">f</font> being  <a href="pua2mss1.html#NM3">operation</a> of <font color="Maroon">A</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">f</font> iff <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="pua2mss1.html#NM3">operation</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<a NAME="E1:59_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> <a href="pua2mss1.html#R2" target="_self">is_exactly_partitable_wrt</a> <font color="Maroon">P</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">a</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">a</font>;<br/><b>assume </b><a NAME="E1:59_1_1_1"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:59_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><b>then consider </b><font color="Maroon">pp</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E4:59_1_1_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">pp</font>
 <b>by </b><i/>;<br/><a NAME="E5:59_1_1_1"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">pp</font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E6:59_1_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="card_3.html#K4">product</a> <font color="Maroon">pp</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:59_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">pp</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:59_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:59_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> iff <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:59_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:59_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,0
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1"><i><font color="Green">Lemma22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:59"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:59_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">EP</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a><br/>

<b>assume </b><a NAME="E2:59_2"/><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EP</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> =  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EP</font>,<font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font> <b>by </b><i><a class="txt" href="pua2mss1.html#E10:59"><i><font color="Green">E36</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be   <a href="pua2mss1.html#NM3">operation</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>assume </b><a NAME="E1:59_2_1"/><i><font color="Green">E78</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 ;<br/><a NAME="E2:59_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><b>then consider </b><font color="Maroon">pp</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E4:59_2_1"/><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">pp</font>
 <b>by </b><i/>;<br/><a NAME="E5:59_2_1"/>
<font color="Maroon">f</font> <a href="pua2mss1.html#R2" target="_self">is_exactly_partitable_wrt</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><a NAME="E6:59_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="pua2mss1.html#R1" target="_self">is_partitable_wrt</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E8:59_2_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">pp</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E2:59"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E9:59_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/><a NAME="E10:59_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">pp</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:59_2_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">pp</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">pp</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E12:59_2_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font> &amp;  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EP</font>,<font color="Olive">x</font> ) )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E10:59"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><a NAME="E13:59_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EP</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T30">EQREL_1:30</a></i>;<br/><b>hence </b><a NAME="E14:59_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:59_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="pua2mss1.html#K7" target="_self">|^</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a>, , </i>;<br/>
<b>hence </b><a NAME="E7:59_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E12:59"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:59_3"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">aa</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EP</font>,<font color="Olive">x</font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E10:59"><i><font color="Green">E36</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><a NAME="E2:59_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EP</font> &amp; <font color="Maroon">EP</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a>, <a class="ref" href="eqrel_1.html#T30">EQREL_1:30</a></i>;<br/><b>hence </b><a NAME="E3:59_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E14:59"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:59"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/>
<b>hence </b><a NAME="E16:59"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>


</div>
