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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:25_1"/><i><font color="Green">E13</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:25_1"/><i><font color="Green">E14</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:25_1"/><i><font color="Green">E15</font></i>: 
 <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="yellow15.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:25_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E7:25_1"/><i><font color="Green">E16</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E8:25_1"/><i><font color="Green">E17</font></i>: 
 <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="yellow15.html#D2" target="_self">Def2</a></i>;<br/><b>assume </b><a NAME="E9:25_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E11:25_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E12:25_1"/><i><font color="Green">E19</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow15.html#E4:25_1"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E13:25_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="yellow15.html#E2:25_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow15.html#E3:25_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>now </b></a><div class="add"><a NAME="E1:25_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow15.html#E1:25"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow15.html#E3:25_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow15.html#E7:25_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E2:25_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow15.html#E12:25_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:25_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>, <a href="margrel1.html#K6">BOOLEAN</a> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T30">FINSEQ_2:30</a></i>;<br/><a NAME="E4:25_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>, <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/><a NAME="E5:25_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>, <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow15.html#E13:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/><a NAME="E6:25_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow15.html#E1:25_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/><b>hence </b><a NAME="E7:25_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> 
 <b>by </b><i><a class="txt" href="yellow15.html#E3:25_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E17:25_1"/><i><font color="Green">E21</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="yellow15.html#E1:25"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow15.html#E3:25_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow15.html#E7:25_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow15.html#E13:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/> <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="yellow15.html#E13:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="yellow15.html#E12:25_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
;<br/><a NAME="E19:25_1"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/> <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow15.html#E19:25_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="yellow15.html#E17:25_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
;<br/><a NAME="E21:25_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>hence </b><a NAME="E22:25_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow15.html#E8:25_1"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E23:25_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</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="E1:25_1_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:25_1_4"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:25_1_4"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E5:25_1_4"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="yellow15.html#E3:25_1_4"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow15.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E6:25_1_4"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow15.html#E17:25_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E7:25_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:25_1_4"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/>
<a NAME="E9:25_1_4"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="yellow15.html#E13:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<i><font color="Green">E29</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="yellow15.html#E6:25_1_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
;<br/>
<a NAME="E12:25_1_4"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>, <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="yellow15.html#E1:25"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow15.html#E3:25_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow15.html#E7:25_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E13:25_1_4"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="yellow15.html#E13:25_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow15.html#E8:25_1_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_4_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_4_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_4_2_1_1"><b>suppose </b></a><a NAME="E1:25_1_4_2_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="yellow15.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="yellow15.html#E6:25_1_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="yellow15.html#E13:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="yellow15.html#E12:25_1_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="yellow15.html#E4:25_1_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow15.html#E11:25_1_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="yellow15.html#E1:25_1_4_2_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow15.html#T6" target="_self">Th6</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_4_2_1_2"><b>suppose </b></a><a NAME="E1:25_1_4_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:25_1_4_2_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow15.html#E13:25_1_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="yellow15.html#T7" target="_self">Th7</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow15.html#E6:25_1_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow15.html#E13:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow15.html#E12:25_1_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="yellow15.html#E4:25_1_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow15.html#E5:25_1_4"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow15.html#E11:25_1_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="yellow15.html#E2:25_1_4_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow15.html#T7" target="_self">Th7</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E15:25_1_4"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow15.html#E9:25_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow15.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E16:25_1_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow15.html#E14:25_1_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E24:25_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="yellow15.html#E11:25_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="setfam_1.html#T59">SETFAM_1:59</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
 <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">c<sub>2</sub></font> <a href="setfam_1.html#R1">is_finer_than</a>  <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/>


</div>
