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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be    <a href="pua2mss1.html#M3" target="_self">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:59"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></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">c<sub>7</sub></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E10:59"/><i><font color="Green">E37</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">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></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">E38</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>15</sub></font> =  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pua2mss1.html#E2:59"><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">E39</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E1:59_1"><i><font color="Green">E38</font></i></a>, <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">b<sub>1</sub></font> being  <a href="pua2mss1.html#NM3">operation</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></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">c<sub>13</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> iff <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></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">c<sub>14</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></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">c<sub>16</sub></font> be   <a href="pua2mss1.html#NM3">operation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<a NAME="E1:59_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="pua2mss1.html#R2" target="_self">is_exactly_partitable_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D11" target="_self">Def11</a></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">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> be    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>15</sub></font>;<br/><b>assume </b><a NAME="E1:59_1_1_1"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>21</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E2:59_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>21</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>20</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>23</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:59_1_1_1"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>21</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E5:59_1_1_1"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:59_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pua2mss1.html#E4:59_1_1_1"><i><font color="Green">E42</font></i></a>, <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">E43</font></i>: 
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:59_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pua2mss1.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E7:59_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>22</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E4:59_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pua2mss1.html#T27" target="_self">Th27</a></i>;<br/><b>hence </b><a NAME="E8:59_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>22</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E6:59_1_1_1"><i><font color="Green">E43</font></i></a></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">c<sub>17</sub></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">c<sub>13</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> iff <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></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">c<sub>14</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E4:59_1"><i><font color="Green">E39</font></i></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D5" target="_self">Def5</a></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,0
 <b>by </b><i><a class="ref" href="pua2mss1.html#T17" target="_self">Th17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:59"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></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">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>16</sub></font> =  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pua2mss1.html#E2:59"><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">c<sub>17</sub></font> be   <a href="pua2mss1.html#NM3">operation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>assume </b><a NAME="E1:59_2_1"/><i><font color="Green">E41</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>14</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>15</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E2:59_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>14</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>20</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:59_2_1"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>14</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E5:59_2_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="pua2mss1.html#R2" target="_self">is_exactly_partitable_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E6:59_2_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="pua2mss1.html#R1" target="_self">is_partitable_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#D10" target="_self">Def10</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:59_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>17</sub></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">c<sub>20</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E9:59_2_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E2:59_2"><i><font color="Green">E40</font></i></a>, <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">c<sub>18</sub></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">c<sub>15</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E4:59_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pua2mss1.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E11:59_2_1"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>14</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></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">c<sub>20</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>15</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></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">c<sub>20</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:59_2_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pua2mss1.html#E4:59_2_1"><i><font color="Green">E42</font></i></a>, <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">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>14</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>15</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E2:59"><i><font color="Green">E36</font></i></a>, <a class="txt" href="pua2mss1.html#E8:59_2_1"><i><font color="Green">E43</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">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>14</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>15</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></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">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>14</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>15</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></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">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:59_2"><i><font color="Green">E39</font></i></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">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="pua2mss1.html#K7" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E1:59_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="pua2mss1.html#E2:59_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pua2mss1.html#D6" target="_self">Def6</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">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T18" target="_self">Th18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:59"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="pua2mss1.html#E10:59"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pua2mss1.html#E11:59"><i><font color="Green">E38</font></i></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">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:59_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E2:59"><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">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E12:59"><i><font color="Green">E39</font></i></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E15:59"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></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">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>


</div>
