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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="rearran1.html#NM1">RearrangmentGen</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="rfinseq.html#K2">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="rfunct_3.html#K16">CHI</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:36"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rearran1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E3:36"/><i><font color="Green">E34</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K16">CHI</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K16">CHI</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rearran1.html#T12" target="_self">Th12</a>, <a class="ref" href="rfinseq.html#D3">RFINSEQ:def 3</a>, <a class="ref" href="rfunct_3.html#D6">RFUNCT_3:def 6</a></i>;<br/>
<a NAME="E4:36"/><i><font color="Green">E35</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rearran1.html#T21" target="_self">Th21</a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finset_1.html#T29">FINSET_1:29</a></i>;<br/>
<a NAME="E6:36"/><i><font color="Green">E36</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</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"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</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"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1"/>
(  not <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="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> or <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="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_1"/>
not <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="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rearran1.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_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="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</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="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> );<br/><a NAME="E2:36_1_1_1_2"/><i><font color="Green">E38</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E3:36_1_1_1_2"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>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="E5:36_1_1_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green">E38</font></i></a>);<br/></i><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:36_1_1_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green">E38</font></i></a>);<br/></i><a NAME="E8:36_1_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E9:36_1_1_1_2"/><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:36_1_1_1_2"/><b>then </b>
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T4">FINSEQ_2:4</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="finseq_2.html#T5">FINSEQ_2:5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E13:36_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:36_1_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:36_1_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E16:36_1_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:36_1_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>13</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E18:36_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</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="E1:36_1_1_1_2_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:36_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>13</sub></font> &amp; not <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_1_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E4:36_1_1_1_2_1_1"/><i><font color="Green">E49</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; ( <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 or <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> ) )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:36_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:36_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:36_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:36_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>17</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="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_1_1_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E8:36_1_1_1_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_1_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E6:36_1_1_1_2_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rearran1.html#E8:36_1_1_1_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><a NAME="E4:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2_1_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:36_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:36_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>16</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="E2:36_1_1_1_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:36_1_1_1_2_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>16</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="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E5:36_1_1_1_2_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:36_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E7:36_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:36_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E9:36_1_1_1_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:36_1_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:36_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E12:36_1_1_1_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then </b><i><font color="Green">E47</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E17:36_1_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_1_1_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</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="E1:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:36_1_1_1_2_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E14"><i><font color="Green">Lemma14</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1_1_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> );<br/>
<a NAME="E4:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>18</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>18</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<a NAME="E3:36_1_1_1_2_3_1_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:36_1_1_1_2_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E5:36_1_1_1_2_3_1_1_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rearran1.html#D2" target="_self">Def2</a></i>;<br/><b>then </b><i><font color="Green">E57</font></i>: <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> )
 </span><b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_1_1_1_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green">E52</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:36_1_1_1_2_3_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_1_1_1"><i><font color="Green">E51</font></i></a>);<br/></i>
<a NAME="E8:36_1_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rearran1.html#E6:36_1_1_1_2_3_1_1_1_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:36_1_1_1_2_3_1_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>16</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="E2:36_1_1_1_2_3_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>assume </b><a NAME="E4:36_1_1_1_2_3_1_1_1"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E5:36_1_1_1_2_3_1_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E7:36_1_1_1_2_3_1_1_1"/><i><font color="Green">E52</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>18</sub></font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>

<a NAME="E1:36_1_1_1_2_3_1_1_1_2"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rearran1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:36_1_1_1_2_3_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rearran1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:36_1_1_1_2_3_1_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E4:36_1_1_1_2_3_1_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_1_1_2_3_1_1_1_2"/>
<font color="Maroon">c<sub>18</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_1_1_2_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_1_1"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1_1_2"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:36_1_1_1_2_3_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>18</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="rearran1.html#E7:36_1_1_1_2_3_1_1_1"><i><font color="Green">E52</font></i></a>);<br/></i>
<a NAME="E10:36_1_1_1_2_3_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:36_1_1_1_2_3_1_1_1"/><b>then </b>
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T4">FINSEQ_2:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="finseq_2.html#T5">FINSEQ_2:5</a></i>;<br/>
<a NAME="E13:36_1_1_1_2_3_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E14:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E15:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>19</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E17:36_1_1_1_2_3_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>18</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="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rearran1.html#E15:36_1_1_1_2_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E6:36_1_1_1_2_3_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rearran1.html#E17:36_1_1_1_2_3_1_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1_2_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_1_1_3_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_3_1_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_3_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_3_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E4:36_1_1_1_2_3_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rearran1.html#E15:36_1_1_1_2_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_2_3_1_1_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E6:36_1_1_1_2_3_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rearran1.html#E17:36_1_1_1_2_3_1_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E19:36_1_1_1_2_3_1_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:36_1_1_1_2_3_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rearran1.html#E19:36_1_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>14</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="rearran1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E14"><i><font color="Green">Lemma14</font></i></a></i>;<br/><a NAME="E7:36_1_1_1_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E9:36_1_1_1_2_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_1_1_1_2_3_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</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="E1:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:36_1_1_1_2_3_1_2_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:36_1_1_1_2_3_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E14"><i><font color="Green">Lemma14</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> );<br/>
<a NAME="E5:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>20</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>20</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<a NAME="E3:36_1_1_1_2_3_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:36_1_1_1_2_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_1_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>20</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rearran1.html#T2" target="_self">Th2</a></i>;<br/><b>then </b><i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_2_1_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green">E56</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:36_1_1_1_2_3_1_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="rearran1.html#E6:36_1_1_1_2_3_1_2_1_1"><i><font color="Green">E55</font></i></a>);<br/></i>
<a NAME="E9:36_1_1_1_2_3_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2_3_1_2_1_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:36_1_1_1_2_3_1_2_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>18</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="E2:36_1_1_1_2_3_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E4:36_1_1_1_2_3_1_2_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E5:36_1_1_1_2_3_1_2_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>assume </b><a NAME="E6:36_1_1_1_2_3_1_2_1"/><i><font color="Green">E54</font></i>: 
not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1"/>
(  not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 </span><b>by </b><i><a class="txt" href="rearran1.html#E6:36_1_1_1_2_3_1_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green">E55</font></i>: 
not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green">E56</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>20</sub></font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>

<a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rearran1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rearran1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_1_1"/>
<font color="Maroon">c<sub>20</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_2_1_1_1"><i><font color="Green">E57</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_2_1_2_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_2_1_1_1"><i><font color="Green">E57</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>20</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E56</font></i></a>);<br/></i><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_2_1_1"/><b>then </b>
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T4">FINSEQ_2:4</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="finseq_2.html#T5">FINSEQ_2:5</a></i>;<br/><a NAME="E8:36_1_1_1_2_3_1_2_1_2_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:36_1_1_1_2_3_1_2_1_2_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E10:36_1_1_1_2_3_1_2_1_2_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:36_1_1_1_2_3_1_2_1_2_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>21</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>20</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="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>21</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rearran1.html#E10:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rearran1.html#E12:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rearran1.html#E11:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rearran1.html#E10:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rearran1.html#E12:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:36_1_1_1_2_3_1_2_1_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green">E56</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E55</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>20</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E56</font></i></a>);<br/></i><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green">E58</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_2_1_2"/><b>then </b>
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T4">FINSEQ_2:4</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="finseq_2.html#T5">FINSEQ_2:5</a></i>;<br/><a NAME="E8:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>20</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="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 1
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( 1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>21</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="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E7:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E8:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E9:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rearran1.html#E8:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E10:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E12:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E13:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:36_1_1_1_2_3_1_2_1_2_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:36_1_1_1_2_3_1_2_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>hence </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rearran1.html#E19:36_1_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E21:36_1_1_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:36_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:36"/>
 <a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="ref" href="rfinseq.html#D1">RFINSEQ:def 1</a></i>;<br/>


</div>
