<?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>set </b><font color="Maroon">c<sub>2</sub></font> =  <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>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pua2mss1.html#NM3">operation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D9" target="_self">PUA2MSS1:def 9</a>,<a class="ref" href="pua2mss1.html#D10" target="_self">PUA2MSS1:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <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>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <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>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1"/>
(  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:53_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:53_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <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>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <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>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b><a NAME="E6:53_1_1_1"/>
<font color="Maroon">c<sub>3</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>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:53_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</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>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:53_1_1_1_1"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T10">FUNCT_6:10</a></i>;<br/>
<a NAME="E5:53_1_1_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font> ) &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:53_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pua2mss1.html#E3:53_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E7:53_1_1_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<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="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="txt" href="pua2mss1.html#E5:53_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</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>;<br/>
<a NAME="E9:53_1_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E5:53_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pua2mss1.html#E7:53_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:53_1_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font>
 
;<br/>
<a NAME="E11:53_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font>]
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:53_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pua2mss1.html#E4:53_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<a NAME="E12:53_1_1_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Olive">b<sub>3</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Olive">b<sub>4</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:53_1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</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> )
 ;<br/>

<a NAME="E2:53_1_1_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></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#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E3:53_1_1_1_1_1"/>
 <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</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#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E4:53_1_1_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>17</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>18</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; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</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="txt" href="pua2mss1.html#E1:53_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:53_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pua2mss1.html#D5" target="_self">Def5</a></i>;<br/>

<a NAME="E6:53_1_1_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E1:53_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:53_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</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>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:53_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="pua2mss1.html#E4:53_1_1_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pua2mss1.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E2:53_1_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</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>19</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="ref" href="pua2mss1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:53_1_1_1_1_1_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>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</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>19</sub></font> &amp; <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>19</sub></font> is <a href="partfun1.html#V1">total</a> &amp; <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>19</sub></font> is <a href="relat_2.html#V3">symmetric</a> &amp; <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>19</sub></font> is <a href="relat_2.html#V8">transitive</a> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:53_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="pua2mss1.html#E4:53_1_1_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="pua2mss1.html#E5:53_1_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="pua2mss1.html#D6" target="_self">Def6</a>, <a class="ref" href="pua2mss1.html#T22" target="_self">Th22</a></i>;<br/><b>hence </b><a NAME="E4:53_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</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>19</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:53_1_1_1_1_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>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span></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="txt" href="pua2mss1.html#E6:53_1_1_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="pua2mss1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E9:53_1_1_1_1_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>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></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="txt" href="pua2mss1.html#E2:53_1_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/>
<b>hence </b><a NAME="E10:53_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</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>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:53_1_1_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:53_1_1_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:53_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</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="txt" href="pua2mss1.html#E5:53_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <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> ;<br/>
<a NAME="E4:53_1_1_1_1_2"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) ) &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E3:53_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pua2mss1.html#E3:53_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E5:53_1_1_1_1_2"/><b>then </b>
( 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>16</sub></font> <a href="hidden.html#R1">=</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="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <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#E1:53_1_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:53_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></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#T30">EQREL_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:53_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>]
 
<b>from </b><i><a class="ref" href="pua2mss1.html#S4" target="_self">PUA2MSS1:sch 4</a>(<a class="txt" href="pua2mss1.html#E11:53_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pua2mss1.html#E10:53_1_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pua2mss1.html#E12:53_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="pua2mss1.html#E13:53_1_1_1_1"><i><font color="Green">E40</font></i></a>);<br/></i>
<b>hence </b><a NAME="E15:53_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:53_1_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_2"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:53_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>then </b><font color="Maroon">c<sub>3</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>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="relat_1.html#T145">RELAT_1:145</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>
;<br/><a NAME="E4:53_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</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>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E5:53_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <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> st <font color="Maroon">c<sub>3</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>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <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>;<br/>

<b>assume </b><a NAME="E2:53"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:53"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</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="E5:53"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E6:53"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T10">FUNCT_6:10</a></i>;<br/>
<a NAME="E7:53"/><b>then </b><i><font color="Green">E34</font></i>: 
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font> ) &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E4:53"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pua2mss1.html#E5:53"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E9:53"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="txt" href="pua2mss1.html#E7:53"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</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>;<br/>
<a NAME="E11:53"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E7:53"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pua2mss1.html#E9:53"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:53"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<a NAME="E13:53"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>]
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E4:53"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E14:53"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Olive">b<sub>3</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Olive">b<sub>4</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><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="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:53_2"/><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:53_2"/>
 <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</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#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E3:53_2"/><b>then </b>
( <font color="Maroon">c<sub>13</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>14</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; <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="txt" href="pua2mss1.html#E1:53_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E4:53_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:53_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pua2mss1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:53"/><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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><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="E1:53_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:53_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</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="txt" href="pua2mss1.html#E7:53"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <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> ;<br/>
<a NAME="E4:53_3"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) ) &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E4:53"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pua2mss1.html#E5:53"><i><font color="Green">E33</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E5:53_3"/><b>then </b>
( 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>12</sub></font> <a href="hidden.html#R1">=</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="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E1:53_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:53_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></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#T30">EQREL_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:53"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>]
 
<b>from </b><i><a class="ref" href="pua2mss1.html#S4" target="_self">PUA2MSS1:sch 4</a>(<a class="txt" href="pua2mss1.html#E13:53"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pua2mss1.html#E12:53"><i><font color="Green">E36</font></i></a>, <a class="txt" href="pua2mss1.html#E14:53"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pua2mss1.html#E15:53"><i><font color="Green">E39</font></i></a>);<br/></i>
<b>hence </b><a NAME="E17:53"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div>
