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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:47"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<b>thus </b><a NAME="E3:47"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:47_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
  <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="47_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:47_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:47_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T132">FINSEQ_2:132</a></i>;<br/>
<a NAME="E3:47_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:47_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">Y</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">Y</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E6:47_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 ;<br/>
<a NAME="E7:47_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E8:47_1_1"/>
<font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#D2">FINSEQ_2:def 2</a></i>;<br/>
<a NAME="E9:47_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</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="E10:47_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:47_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">a</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="E2:47_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E3:47_1"/><b>then </b>
<font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T132">FINSEQ_2:132</a></i>;<br/>
<a NAME="E4:47_1"/><b>then </b>
<font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:47_1"/><b>then </b>
<font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E7:47_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/>
<a NAME="E8:47_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E9:47_1"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E10:47_1"/>
<font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#D2">FINSEQ_2:def 2</a></i>;<br/>
<a NAME="E11:47_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</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="E12:47_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:47_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="circcmb3.html#T1" target="_self">Th1</a></i>;<br/>


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

<b>hence </b><a NAME="E4:47"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T130">FINSEQ_2:130</a></i>;<br/>


</div>
