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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>given </b><font color="Maroon">c<sub>5</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><a NAME="E1:38_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E2:38_1_2"/><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3" target="_self">Carrier</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:38_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<b>given </b><font color="Maroon">c<sub>6</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><a NAME="E4:38_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:38_1_2"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3" target="_self">Carrier</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E6:38_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E7:38_1_2"/><i><font color="Green">E29</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="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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="txt" href="rlvect_2.html#E1:38_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="rlvect_2.html#E2:38_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="rlvect_2.html#E4:38_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="rlvect_2.html#E5:38_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rlvect_2.html#D9" target="_self">Def9</a>, <a class="ref" href="finseq_1.html#T65">FINSEQ_1:65</a></i>;<br/>
<a NAME="E8:38_1_2"/><i><font color="Green">E30</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</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>5</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</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>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:38_1_2"/><i><font color="Green">E31</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<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 <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E10:38_1_2"/><i><font color="Green">E32</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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> 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="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</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="38_1_2_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> ;<br/>

<b>assume </b><a NAME="E1:38_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:38_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E2:38_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="rlvect_2.html#E5:38_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="rlvect_2.html#E7:38_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_2.html#E8:38_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<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="E4:38_1_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E1:38_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T144">FUNCT_1:144</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<a NAME="E5:38_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E4:38_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:38_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>

<b>thus </b><a NAME="E7:38_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>]
 <b>by </b><i><a class="txt" href="rlvect_2.html#E4:38_1_2_1"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:38_1_2"/><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>, <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E13:38_1_2"/><i><font color="Green">E34</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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</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>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="rlvect_2.html#E10:38_1_2"><i><font color="Green">E32</font></i></a>);<br/></i>
<a NAME="E14:38_1_2"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_1_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>10</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="E2:38_1_2_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E3:38_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E2:38_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="rlvect_2.html#E5:38_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:38_1_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E6:38_1_2_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</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>5</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#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:38_1_2_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E7:38_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_2.html#E8:38_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rlvect_2.html#E5:38_1_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:38_1_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E2:38_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rlvect_2.html#E6:38_1_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>
<a NAME="E9:38_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:38_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/>
<a NAME="E10:38_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</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><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E7:38_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_2.html#E8:38_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rlvect_2.html#E13:38_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rlvect_2.html#E5:38_1_2_2"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E11:38_1_2_2"/><b>then </b>
<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#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T24">ZFMISC_1:24</a></i>;<br/>
<b>hence </b><a NAME="E12:38_1_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E7:38_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:38_1_2"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:38_1_2_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E2:38_1_2_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E3:38_1_2_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E11:38_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rlvect_2.html#E1:38_1_2_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:38_1_2_3"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</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>10</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</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><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E13:38_1_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:38_1_2_3"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</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="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E2:38_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="rlvect_2.html#E5:38_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="rlvect_2.html#E7:38_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_2.html#E8:38_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rlvect_2.html#E3:38_1_2_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:38_1_2_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:38_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E2:38_1_2_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="rlvect_2.html#E4:38_1_2_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T161">FUNCT_1:161</a></i>;<br/>
<a NAME="E8:38_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <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>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E7:38_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_2.html#E8:38_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rlvect_2.html#E11:38_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rlvect_2.html#E1:38_1_2_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/>
<b>hence </b><a NAME="E9:38_1_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E4:38_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="rlvect_2.html#E14:38_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/>
<a NAME="E17:38_1_2"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</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>5</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></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>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <b>by </b><i><a class="txt" href="rlvect_2.html#E7:38_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_4"><b>now </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="E1:38_1_2_4"/><i><font color="Green">E36</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"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rlvect_2.html#E7:38_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_2.html#E9:38_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rlvect_1.html#T54">RLVECT_1:54</a></i>;<br/><a NAME="E3:38_1_2_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E7:38_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_2.html#E9:38_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rlvect_2.html#E1:38_1_2_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:38_1_2_4"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>11</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="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E14:38_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>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="rlvect_2.html#E8:38_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rlvect_2.html#E8:38_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rlvect_2.html#E4:38_1_2_4"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rlvect_1.html#T54">RLVECT_1:54</a></i>;<br/><span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="rlvect_2.html#E4:38_1_2_4"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<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></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_2.html#E7:38_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_2.html#E8:38_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rlvect_2.html#E9:38_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rlvect_2.html#E13:38_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rlvect_2.html#E1:38_1_2_4"><i><font color="Green">E36</font></i></a></i>
;<br/><a NAME="E8:38_1_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<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></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T145">FUNCT_1:145</a></i>;<br/><a NAME="E9:38_1_2_4"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" 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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <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>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="rlvect_2.html#E7:38_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_2.html#E8:38_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rlvect_2.html#E9:38_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rlvect_2.html#E1:38_1_2_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rlvect_2.html#E4:38_1_2_4"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rlvect_2.html#D9" target="_self">Def9</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="ref" href="zfmisc_1.html#T24">ZFMISC_1:24</a></i>;<br/><b>hence </b><a NAME="E10:38_1_2_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" 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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5" target="_self">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:38_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E3:38_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="rlvect_2.html#E6:38_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="rlvect_2.html#E7:38_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rlvect_2.html#T8" target="_self">Th8</a></i>;<br/>


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