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

<b>let </b><font color="Maroon">f</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">A</font> =  <a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font>;<br/>
<a NAME="E1:17"/><i><font color="Green">E32</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lang1.html#K5">Terminals</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="lang1.html#K6">NonTerminals</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span></span>)</span> &amp;  <a href="lang1.html#K5">Terminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="lang1.html#K6">NonTerminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="dtconstr.html#T8">DTCONSTR:8</a>, <a class="ref" href="lang1.html#T1">LANG1:1</a></i>;<br/>
<a NAME="E2:17"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="lang1.html#K5">Terminals</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="lang1.html#K6">NonTerminals</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>thus </b><a NAME="E3:17"/>
 <a href="lang1.html#K5">Terminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_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="E1:17_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span>
 ;<br/>

<a NAME="E2:17_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xd</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font> <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xa</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:17_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">fx</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">xd</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="finseq_2.html#T77">FINSEQ_2:77</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E6:17_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xd</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><a NAME="E7:17_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xd</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="freealg.html#K2" target="_self">REL</a> <font color="Maroon">f</font>,<font color="Maroon">X</font> &amp; <font color="Maroon">xd</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E8:17_1_1"/><b>then </b>
<font color="Maroon">xa</font> <a href="lang1.html#R1">==&gt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/><a NAME="E9:17_1_1"/><b>then </b>
<font color="Maroon">xa</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">n</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">t</font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">n</font> </span>}</span> 
 ;<br/><a NAME="E10:17_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K6">NonTerminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="lang1.html#D3">LANG1:def 3</a></i>;<br/><b>hence </b><a NAME="E11:17_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E4:17"/>
 <a href="lang1.html#K6">NonTerminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</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="17_2"><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="E1:17_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K6">NonTerminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span>
 ;<br/>

<a NAME="E2:17_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">n</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">t</font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">n</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="lang1.html#D3">LANG1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">t</font> being   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E4:17_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  ex <font color="Olive">n</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Maroon">t</font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">n</font> )
 ;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E6:17_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">t</font> <a href="lang1.html#R1">==&gt;</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:17_2"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="lang1.html#U1">Rules</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:17_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="freealg.html#K2" target="_self">REL</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:17_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/>


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

<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="E5:17"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">xa</font> = <font color="Maroon">x</font> as   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xd</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:17"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">fx</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">xd</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="finseq_2.html#T77">FINSEQ_2:77</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E12:17"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xd</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E13:17"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xd</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="freealg.html#K2" target="_self">REL</a> <font color="Maroon">f</font>,<font color="Maroon">X</font> &amp; <font color="Maroon">xd</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:17"/><b>then </b>
<font color="Maroon">xa</font> <a href="lang1.html#R1">==&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<a NAME="E15:17"/><b>then </b>
<font color="Maroon">xa</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">n</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">t</font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">n</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E16:17"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K6">NonTerminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="lang1.html#D3">LANG1:def 3</a></i>;<br/>


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