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

<b>set </b><font color="Maroon">fgs</font> =  <a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">fua</font> =  <a href="freealg.html#K15" target="_self">FreeUnivAlgZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>;<br/>
<a NAME="E1:42_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" target="_self">FreeUnivAlgZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E2:42_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" target="_self">FreeUnivAlgZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="freealg.html#D5" target="_self">FREEALG:def 5</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">D</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="trees_2.html#NM6">DecoratedTree</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">D</font></span>)</span>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:42_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">s</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">D</font></span>)</span>  st <font color="Olive">s</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">D</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Olive">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <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/>

<b>assume </b><a NAME="E1:42_1_1_1"/>
<font color="Maroon">s</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">D</font></span>)</span>
 ;<br/>

<a NAME="E2:42_1_1_1"/><b>then </b>
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">q</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">D</font></span>)</span> : <font color="Olive">q</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">D</font></span>)</span> </span>}</span> 
 
;<br/>
<a NAME="E3:42_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
 
;<br/>
<a NAME="E4:42_1_1_1"/>
 <a href="unialg_2.html#K6">Constants</a> <span class="p1">(<span class="default"><a href="freealg.html#K15" target="_self">FreeUnivAlgZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:42_1_1_1"/><b>then </b>
 <a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D13">UNIALG_2:def 13</a></i>;<br/>
<b>hence </b><a NAME="E6:42_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">s</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:42_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">nt</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">D</font></span>)</span><br/> for <font color="Olive">ts</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">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Olive">ts</font> &amp; ( for <font color="Olive">t</font> being  <a href="trees_2.html#NM6">DecoratedTree</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">D</font></span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">ts</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <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/><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><b>that </b><br/><a NAME="E1:42_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">s</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:42_1_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">t</font> being  <a href="trees_2.html#NM6">DecoratedTree</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">D</font></span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> holds <br/><font color="Olive">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">B</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" target="_self">FreeUnivAlgZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E4:42_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">B</font> is <a href="unialg_2.html#V1">opers_closed</a>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E5:42_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K16" target="_self">FreeGenSetZAO</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="42_1_1_2_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="E1:42_1_1_2_1"/><i><font color="Green">E57</font></i>: 
<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/>

<a NAME="E2:42_1_1_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="trees_3.html#M5">Element</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> <b>by </b><i/>;<br/>
<a NAME="E4:42_1_1_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:42_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">cp</font> = <font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E7:42_1_1_2"/><i><font color="Green">E58</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">p</font></span>)</span></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">D</font></span>)</span> &amp; 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">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="freealg.html#K2" target="_self">REL</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s0</font> = <font color="Maroon">s</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">rp</font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">p</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:42_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s0</font>,<font color="Maroon">rp</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>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<a NAME="E11:42_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">s0</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s0</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">rp</font> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">ns</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E13:42_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="freealg.html#K14" target="_self">FreeOpSeqZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>then </b> <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="freealg.html#K14" target="_self">FreeOpSeqZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> = 
 <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">f</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E15:42_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="freealg.html#K14" target="_self">FreeOpSeqZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ns</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="freealg.html#K14" target="_self">FreeOpSeqZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="freealg.html#K14" target="_self">FreeOpSeqZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ns</font> <a href="hidden.html#R1">=</a>  <a href="freealg.html#K13" target="_self">FreeOpZAO</a> <font color="Maroon">ns</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" target="_self">FreeUnivAlgZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K1">Operations</a> <span class="p1">(<span class="default"><a href="freealg.html#K15" target="_self">FreeUnivAlgZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="freealg.html#D1" target="_self">Def1</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o</font> =  <a href="freealg.html#K13" target="_self">FreeOpZAO</a> <font color="Maroon">ns</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font> as   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" target="_self">FreeUnivAlgZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> ;<br/>
<a NAME="E17:42_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">B</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">o</font>
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#D5">UNIALG_2:def 5</a></i>;<br/>
<a NAME="E18:42_1_1_2"/><i><font color="Green">E63</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="freealg.html#K13" target="_self">FreeOpZAO</a> <font color="Maroon">ns</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <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">ns</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><a class="ref" href="freealg.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">tp</font> = <font color="Maroon">p</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 href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">p</font></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>
<br/>.= 
 <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">p</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>then </b><i><font color="Green">E64</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">rp</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ns</font>
<b>by </b><i><a class="ref" href="freealg.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E22:42_1_1_2"/><b>then </b>
<font color="Maroon">tp</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</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">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ns</font> </span>}</span> 
 
;<br/>
<a NAME="E23:42_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">cp</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E3:42_1_1"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E24:42_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" target="_self">FreeUnivAlgZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/>
<a NAME="E25:42_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ns</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E3:42_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pralg_1.html#T1">PRALG_1:1</a></i>;<br/>
<font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">cp</font> = 
<span class="p1">(<span class="default"><a href="freealg.html#K4" target="_self">Sym</a> <font color="Maroon">ns</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><a class="ref" href="freealg.html#D1" target="_self">Def1</a>, <a class="ref" href="freealg.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<font color="Maroon">s</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="freealg.html#D1" target="_self">Def1</a>, </i>
;<br/>
<b>hence </b><a NAME="E27:42_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">p</font>]
 <b>by </b><i>, <a class="txt" href="freealg.html#E2"><i><font color="Green">Lemma33</font></i></a>, , <a class="ref" href="unialg_2.html#D4">UNIALG_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:42_1_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">t</font> being  <a href="trees_2.html#NM6">DecoratedTree</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">D</font></span>)</span>  st <font color="Olive">t</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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7">DTCONSTR:sch 7</a>(, );<br/></i>
<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="E6:42_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" target="_self">FreeUnivAlgZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="trees_3.html#M5">Element</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/>
<a NAME="E8:42_1_1"/>
<font color="Maroon">x1</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/>;<br/>
<b>hence </b><a NAME="E9:42_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K16" target="_self">FreeGenSetZAO</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div>
