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

<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">D</font>;<br/>
<b>set </b><font color="Maroon">i</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">Y</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">D</font>;<br/>
<b>set </b><font color="Maroon">smm</font> =  <a href="freealg.html#K4" target="_self">Sym</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">p</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="freealg.html#K4" target="_self">Sym</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Olive">p</font> ) );<br/>
<a NAME="E1:23_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E2:23_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_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> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:23_1_1_1"/><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="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] )
 ;<br/>

<a NAME="E2:23_1_1_1"/><b>then </b>
<font color="Maroon">x</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 <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <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">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:23_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 ;<br/>
<a NAME="E5:23_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="freealg.html#K4" target="_self">Sym</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">s</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> ;<br/>
<a NAME="E7:23_1_1_1"/>
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> &amp;  <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="dtconstr.html#K1">roots</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">s</font></span>)</span> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<a NAME="E8:23_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sm</font> =  <a href="freealg.html#K4" target="_self">Sym</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</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">D</font> ;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> 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">D</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">rs</font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">s</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">D</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="E12:23_1_1_1"/>
<font color="Maroon">sm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:23_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">sm</font>,<font color="Maroon">rs</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">D</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E14:23_1_1_1"/><b>then </b>
 <a href="freealg.html#K4" target="_self">Sym</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:23_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="dtconstr.html#D4">DTCONSTR:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:23_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y2</font>] )
 ;<br/>

<a NAME="E2:23_1_1_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">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <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">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:23_1_1_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 ;<br/>
<a NAME="E5:23_1_1_2"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="freealg.html#K4" target="_self">Sym</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">s</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="freealg.html#K4" target="_self">Sym</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:23_1_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span><b> such that </b><br/><a NAME="E5:23_1_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  &amp;  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>and </b><br/><a NAME="E6:23_1_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="partfun1.html#S2">PARTFUN1:sch 2</a>(, );<br/></i>
<a NAME="E7:23_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:23_1_1_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</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="23_1_1_3_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="E1:23_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:23_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><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="E2:23_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:23_1_1_3"/><b>then </b>
<font color="Maroon">x</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 <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <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">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E5:23_1_1_3"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">sm</font> =  <a href="freealg.html#K4" target="_self">Sym</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</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">D</font></span>)</span> ;<br/>
<a NAME="E7:23_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">sm</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">s</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:23_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">tu</font> = <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 <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <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">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> </span>}</span> ;<br/>
<a NAME="E8:23_1_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>, <font color="Maroon">y</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:23_1_1_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:23_1_1_4"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x</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 <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <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">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> </span>}</span>  &amp; <font color="Maroon">y</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 <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <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">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> </span>}</span>  )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">sx</font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:23_1_1_4"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">sx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 ;<br/>
<b>consider </b><font color="Maroon">sy</font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E6:23_1_1_4"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E7:23_1_1_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:23_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">y</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>, <font color="Maroon">y</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:23_1_1_5"/><i><font color="Green">E111</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:23_1_1_5"/><b>then </b>
<font color="Maroon">x</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 <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <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">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">sx</font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:23_1_1_5"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">sx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">sy</font> = <font color="Maroon">y</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></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="E6:23_1_1_5"/>
<font color="Maroon">sy</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 <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <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">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:23_1_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <b>by </b><i>, , <a class="ref" href="unialg_1.html#T1">UNIALG_1:1</a>, <a class="ref" href="unialg_1.html#D1">UNIALG_1:def 1</a>, <a class="ref" href="unialg_1.html#D2">UNIALG_1:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E11:23_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">p</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>;<br/>

<b>assume </b><a NAME="E12:23_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>hence </b><a NAME="E13:23_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="freealg.html#K4" target="_self">Sym</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>


</div>
