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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:54"/><i><font color="Green">E43</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="afinsq_1.html#NM1">XFinSequence</a>, <font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K1">^</a> <span class="p1"><a href="afinsq_1.html#K6">&lt;%</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="afinsq_1.html#K6">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T44">AFINSQ_1:44</a>, <a class="txt" href="card_fin.html#E1:54"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E4:54"/><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2"><a href="afinsq_1.html#K6">&lt;%</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="afinsq_1.html#K6">%&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1"><a href="afinsq_1.html#K6">&lt;%</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="afinsq_1.html#K6">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:54"><i><font color="Green">E43</font></i></a>, <a class="txt" href="card_fin.html#E3:54"><i><font color="Green">E44</font></i></a>, <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a>, <a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a></i>;<br/>
<a NAME="E5:54"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:54"><i><font color="Green">E43</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:54"/><b>then </b><i><font color="Green">E46</font></i>: 
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="txt" href="card_fin.html#E48"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<a NAME="E7:54"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="card_fin.html#E1:54_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E3:54_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:54"><i><font color="Green">E44</font></i></a>, <a class="txt" href="card_fin.html#E6:54"><i><font color="Green">E46</font></i></a>, <a class="txt" href="card_fin.html#E1:54_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E4:54_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:54"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:54"><i><font color="Green">E44</font></i></a>, <a class="txt" href="card_fin.html#E6:54"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a>, <a class="ref" href="afinsq_1.html#T40">AFINSQ_1:40</a></i>;<br/>
<b>hence </b><a NAME="E9:54"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal4.html#K1">^</a> <span class="p1"><a href="afinsq_1.html#K6">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="afinsq_1.html#K6">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="card_fin.html#E3:54"><i><font color="Green">E44</font></i></a>, <a class="txt" href="card_fin.html#E4:54"><i><font color="Green">E45</font></i></a></i>;<br/>


</div>
