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

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <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> st <font color="Olive">b<sub>2</sub></font> <a href="alg_1.html#R6" target="_self">is_representatives_FS</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E1:29_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:29_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</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="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:29_1_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <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> such that </b><br/><a NAME="E6:29_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="alg_1.html#R6" target="_self">is_representatives_FS</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="alg_1.html#T19" target="_self">Th19</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>8</sub></font> =  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">c<sub>2</sub></font>,<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>7</sub></font></span>)</span>;<br/>

<i><font color="Green">E26</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</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="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</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="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>
;<br/>
<a NAME="E8:29_1_1_1"/><i><font color="Green">E27</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="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E4:29_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="alg_1.html#E6:29_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="alg_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E9:29_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E10:29_1_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</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="alg_1.html#E7:29_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="alg_1.html#E8:29_1_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E11:29_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <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="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E12:29_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font>
 <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>9</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E13:29_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <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>;<br/>

<b>assume </b><a NAME="E14:29_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="alg_1.html#R6" target="_self">is_representatives_FS</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:29_1_1_1"/><i><font color="Green">E32</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> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E17:29_1_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E4:29_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="alg_1.html#E13:29_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="alg_1.html#E14:29_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="alg_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E18:29_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E19:29_1_1_1"/><b>then </b><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>
 
<b>by </b><i><a class="txt" href="alg_1.html#E7:29_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="alg_1.html#E17:29_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E20:29_1_1_1"/>
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>7</sub></font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:29_1_1_1_2"/><i><font color="Green">E35</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>7</sub></font>
 ;<br/>

<a NAME="E2:29_1_1_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <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="alg_1.html#E6:29_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="alg_1.html#D11" target="_self">Def11</a></i>;<br/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="alg_1.html#E8:29_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="alg_1.html#E17:29_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E4:29_1_1_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <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="alg_1.html#E4:29_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="alg_1.html#E13:29_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="alg_1.html#E14:29_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="alg_1.html#E1:29_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="alg_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E5:29_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</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="alg_1.html#E1:29_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:29_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>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</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>12</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="alg_1.html#E2:29_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="alg_1.html#E4:29_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:29_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</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="alg_1.html#K2" target="_self">ExtendRel</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E8:29_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="alg_1.html#E17:29_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="alg_1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E22:29_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> <font color="Maroon">c<sub>7</sub></font></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>10</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E10:29_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="alg_1.html#E16:29_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="alg_1.html#E19:29_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="alg_1.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E23:29_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">c<sub>2</sub></font>,<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>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="alg_1.html#E11:29_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:29_1_1"/><i><font color="Green">E24</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>(<a class="txt" href="alg_1.html#E1:29_1_1"><i><font color="Green">E23</font></i></a>);<br/></i>
<a NAME="E4:29_1_1"/>
<span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
;<br/>
<a NAME="E5:29_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E6:29_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T128">FINSEQ_2:128</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="alg_1.html#E3:29_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E8:29_1_1"/><i><font color="Green">E25</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</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="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="alg_1.html#E3:29_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/>
<a NAME="E9:29_1_1"/><i><font color="Green">E26</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#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> holds <br/> <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="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:29_1_1_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E3:29_1_1_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="alg_1.html#E8:29_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E5:29_1_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</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="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="alg_1.html#E8:29_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="alg_1.html#E1:29_1_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:29_1_1_2"/>
 <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>8</sub></font>
 <b>by </b><i><a class="txt" href="alg_1.html#E3:29_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="alg_1.html#E5:29_1_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:29_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font> st  <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="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:29_1_1_3"/><i><font color="Green">E27</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>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E3:29_1_1_3"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="alg_1.html#E8:29_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E4:29_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E5:29_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="alg_1.html#E8:29_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="alg_1.html#E1:29_1_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="alg_1.html#E3:29_1_1_3"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="alg_1.html#E3:29_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="alg_1.html#E9:29_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="unialg_1.html#T1">UNIALG_1:1</a>, <a class="ref" href="unialg_1.html#D1">UNIALG_1:def 1</a>, <a class="ref" href="unialg_1.html#D2">UNIALG_1:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E12:29_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="alg_1.html#E3:29_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E13:29_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <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>;<br/>

<b>assume </b><a NAME="E14:29_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="alg_1.html#R6" target="_self">is_representatives_FS</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>hence </b><a NAME="E15:29_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">c<sub>2</sub></font>,<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>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="alg_1.html#E3:29_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="alg_1.html#E13:29_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>


</div>
