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

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="freealg.html#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:29_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#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E2:29_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#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font> =  <a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>] means <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#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E3:29_1_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:29_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:29_1_1_1"/><b>then </b>
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>6</sub></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">b<sub>1</sub></font></span>)</span> where B 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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> </span>}</span> 
 
;<br/>
<a NAME="E3:29_1_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="freealg.html#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E4:29_1_1_1"/>
 <a href="freealg.html#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="freealg.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E5:29_1_1_1"/><b>then </b>
 <a href="freealg.html#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E6:29_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>6</sub></font>]
 <b>by </b><i><a class="txt" href="freealg.html#E3:29_1_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:29_1_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><br/>for <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:29_1_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E2:29_1_1_2"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Olive">b<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#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></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#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E4:29_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></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="E5:29_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></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#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:29_1_1_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:29_1_1_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="freealg.html#E1:29_1_1_2_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E4:29_1_1_2_1"/>
<font color="Maroon">c<sub>10</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#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="freealg.html#E2:29_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="freealg.html#E1:29_1_1_2_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:29_1_1_2_1"/>
<font color="Maroon">c<sub>9</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#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></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#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E7:29_1_1_2"/><i><font color="Green">E21</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="freealg.html#K2" target="_self">REL</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="freealg.html#E1:29_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="txt" href="freealg.html#E7:29_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:29_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:29_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<a NAME="E11:29_1_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="freealg.html#D8" target="_self">Def8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E13:29_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="freealg.html#K6" target="_self">FreeOpSeqNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="freealg.html#D12" target="_self">Def12</a></i>;<br/>
<b>then </b> <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="freealg.html#K6" target="_self">FreeOpSeqNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></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">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E15:29_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="freealg.html#K6" target="_self">FreeOpSeqNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></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#K6" target="_self">FreeOpSeqNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="freealg.html#K6" target="_self">FreeOpSeqNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="freealg.html#K5" target="_self">FreeOpNSG</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="freealg.html#E11:29_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="freealg.html#D12" target="_self">Def12</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="freealg.html#K5" target="_self">FreeOpNSG</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> as   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E17:29_1_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="freealg.html#E4:29_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="unialg_2.html#D5">UNIALG_2:def 5</a></i>;<br/>
<a NAME="E18:29_1_1_2"/><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="freealg.html#K5" target="_self">FreeOpNSG</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="freealg.html#E11:29_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="freealg.html#D11" target="_self">Def11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></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">E25</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="freealg.html#E11:29_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E22:29_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E23:29_1_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="freealg.html#E18:29_1_1_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E24:29_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></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#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/>
<a NAME="E25:29_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="freealg.html#E18:29_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="pralg_1.html#T1">PRALG_1:1</a></i>;<br/>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="freealg.html#K4" target="_self">Sym</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="freealg.html#E11:29_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="freealg.html#E23:29_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="freealg.html#D11" target="_self">Def11</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="freealg.html#E11:29_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="freealg.html#D10" target="_self">Def10</a></i>
;<br/>
<b>hence </b><a NAME="E27:29_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>7</sub></font>]
 <b>by </b><i><a class="txt" href="freealg.html#E17:29_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="freealg.html#E21:29_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="freealg.html#E25:29_1_1_2"><i><font color="Green">E27</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="E5:29_1_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7">DTCONSTR:sch 7</a>(<a class="txt" href="freealg.html#E3:29_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="freealg.html#E4:29_1_1"><i><font color="Green">E17</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>6</sub></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="E6:29_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</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="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E8:29_1_1"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="freealg.html#E6:29_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:29_1_1"/>
<font color="Maroon">c<sub>6</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#K8" target="_self">FreeGenSetNSG</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="freealg.html#E5:29_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div>
