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<b>let </b><font color="Maroon">D</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">r</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:7"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">n</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font> as   <a href="afinsq_1.html#NM1">XFinSequence</a> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1"><b>case </b></a><a NAME="E1:7_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:7_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/><a NAME="E3:7_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:7_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 ;<br/>

<b>hence </b><a NAME="E2:7_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:7_1_1_1"/>
<font color="Maroon">n</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font> is   <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2"><b>case </b></a><a NAME="E1:7_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:7_1_1_2"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="funcop_1.html#T16">FUNCOP_1:16</a></i>;<br/><a NAME="E3:7_1_1_2"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:7_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="afinsq_1.html#T18">AFINSQ_1:18</a></i>;<br/><b>hence </b><a NAME="E5:7_1_1_2"/>
<font color="Maroon">n</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font> is   <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T19">AFINSQ_1:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
<font color="Maroon">n</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font> is   <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">D</font>
 ;<br/>

<b>thus </b><a NAME="E5:7"/>
for <font color="Olive">q2</font> being  <a href="afinsq_1.html#NM1">XFinSequence</a>  st <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font> holds <br/> <a href="afinsq_1.html#K1">len</a> <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q2</font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/>

<b>assume </b><a NAME="E1:7_2"/>
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:7_2"/><b>then </b>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E3:7_2"/>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>


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


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