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<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> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><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="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K18">[*]</a>  )
 ;<br/><a NAME="E2:8_1"/><b>then </b><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D16">FINSEQ_1:def 16</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:8_1"/><i><font color="Green">E10</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E5:8_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</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>5</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>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="taxonom1.html#E1:8_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_1.html#D16">FINSEQ_1:def 16</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E7:8_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K3">Rev</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E8:8_1"/><i><font color="Green">E13</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="taxonom1.html#E4:8_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="taxonom1.html#E7:8_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><a NAME="E9:8_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E4:8_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="taxonom1.html#E7:8_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_5.html#T65">FINSEQ_5:65</a></i>;<br/><i><font color="Green">E15</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</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> = 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</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>
<b>by </b><i><a class="txt" href="taxonom1.html#E7:8_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="taxonom1.html#E4:8_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="taxonom1.html#E7:8_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_5.html#T65">FINSEQ_5:65</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:8_1_2"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E2:8_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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="taxonom1.html#E7:8_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="taxonom1.html#E1:8_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><a NAME="E3:8_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</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="taxonom1.html#E7:8_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="taxonom1.html#E1:8_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><a NAME="E4:8_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</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>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:8_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:8_1_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</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="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:8_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:8_1_2"/><i><font color="Green">E20</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="taxonom1.html#E2:8_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:8_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</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>
 <b>by </b><i><a class="txt" href="taxonom1.html#E6:8_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:8_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <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="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:8_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="taxonom1.html#E2:8_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:8_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</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> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E13:8_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="taxonom1.html#E7:8_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="taxonom1.html#E12:8_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E14:8_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:8_1_2"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E15:8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</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> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E16:8_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></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="taxonom1.html#E5:8_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="taxonom1.html#E12:8_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="taxonom1.html#E13:8_1_2"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E17:8_1_2"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E18:8_1_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>,<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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></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="taxonom1.html#E1:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="taxonom1.html#E16:8_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/><a NAME="E19:8_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="taxonom1.html#E7:8_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="taxonom1.html#E5:8_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_5.html#T61">FINSEQ_5:61</a></i>;<br/><b>hence </b><a NAME="E20:8_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>6</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>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="taxonom1.html#E7:8_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="taxonom1.html#E9:8_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="taxonom1.html#E18:8_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_5.html#T61">FINSEQ_5:61</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:8_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K18">[*]</a> 
 <b>by </b><i><a class="txt" href="taxonom1.html#E2:8_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="taxonom1.html#E8:8_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="taxonom1.html#E9:8_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="taxonom1.html#E10:8_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#D16">FINSEQ_1:def 16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:8"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K18">[*]</a>  <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>


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