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

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



<b>set </b><font color="Maroon">fgs</font> =  <a href="freealg.html#K9" target="_self">FreeGenSetNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">fua</font> =  <a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>;<br/>
<b>let </b><font color="Maroon">U1</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="freealg.html#D6" target="_self">FREEALG:def 6</a><br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green">E32</font></i>: 
 <a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>,<font color="Maroon">U1</font> <a href="unialg_2.html#R1">are_similar</a> 
 ;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="freealg.html#K9" target="_self">FreeGenSetNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>;<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>set </b><font color="Maroon">c1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>;<br/>
<b>set </b><font color="Maroon">cf</font> = 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">f</font>,<font color="Maroon">D</font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <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>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> =  <a href="freealg.html#K10" target="_self">pi</a> <font color="Maroon">F</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <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>,  <a href="finseq_1.html#NM1">FinSequence</a>,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> = <span class="p1">(<span class="default"><a href="freealg.html#K1" target="_self">oper</a> <span class="p2">(<span class="default"><a href="freealg.html#K11" target="_self">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">U1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>3</sub></font>;<br/>
<b>consider </b><font color="Maroon">ff</font> being   <a href="funct_2.html#NM1">Function</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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font><b> such that </b><br/><a NAME="E3:32"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">t</font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>  st <font color="Olive">t</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">f</font>,<font color="Maroon">D</font></span>)</span> holds <br/><font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">t</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>)
 <b>and </b><br/><a NAME="E4:32"/><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> holds <br/><font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">ts</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">nt</font>, <a href="dtconstr.html#K1">roots</a> <font color="Olive">ts</font>,<font color="Maroon">ff</font> <a href="lang1.html#K15">*</a> <font color="Olive">ts</font>)
 <b>from </b><i><a class="ref" href="dtconstr.html#S8">DTCONSTR:sch 8</a>();<br/></i>
<b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">ff</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>,<font color="Maroon">U1</font> ;<br/>
<b>take </b>
<font color="Maroon">ff</font>
;<br/>

<a NAME="E6:32"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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">f</font>,<font color="Maroon">D</font></span>)</span> holds <br/>for <font color="Olive">o1</font> being  <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">f</font>,<font color="Maroon">D</font></span>)</span><br/> for <font color="Olive">o2</font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U1</font>  st <font color="Olive">o1</font> <a href="hidden.html#R1">=</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">f</font>,<font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">o2</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/>for <font color="Olive">x</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</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="Olive">o1</font> holds <br/><font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">o1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">o2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="lang1.html#K15">*</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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">f</font>,<font color="Maroon">D</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">o1</font> be   <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">f</font>,<font color="Maroon">D</font></span>)</span>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U1</font>;<br/>



<b>assume </b><a NAME="E2:32_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</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">f</font>,<font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>;<br/>

<b>assume </b><a NAME="E3:32_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">xa</font> = <font color="Maroon">x</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="E5:32_1"/><i><font color="Green">E57</font></i>: 
(  <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">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> &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">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <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">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <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="freealg.html#K6" target="_self">FreeOpSeqNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> &amp; 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">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</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">f</font>,<font color="Maroon">D</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:32_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">n</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="hidden.html#R1">=</a>  <a href="unialg_1.html#K4">signature</a> <span class="p1">(<span class="default"><a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</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">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/>.= 
<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 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">f</font>,<font color="Maroon">D</font></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>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> </span>}</span> 

;<br/>
<b>then consider </b><font color="Maroon">w</font> being    <a href="finseq_2.html#M2">Element</a> of 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">f</font>,<font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E9:32_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:32_1"/><i><font color="Green">E61</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> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K4">signature</a> <span class="p1">(<span class="default"><a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <span class="p2">(<span class="default"><a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="ref" href="unialg_1.html#D11">UNIALG_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">xa</font></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xa</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">xa</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E12:32_1"/><b>then </b><i><font color="Green">E63</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">xa</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">xa</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">nt</font> = <font color="Maroon">n</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> <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">nd</font> = <font color="Maroon">n</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> <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xa</font> = <font color="Maroon">xa</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">rxa</font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">xa</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D</font> ;<br/>
<b>reconsider </b><font color="Maroon">rxa</font> = <font color="Maroon">rxa</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="E18:32_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">nd</font>,<font color="Maroon">rxa</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="E19:32_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">xa</font>
 
<b>by </b><i><a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<a NAME="E20:32_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">xa</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="freealg.html#K1" target="_self">oper</a> <span class="p2">(<span class="default"><a href="freealg.html#K11" target="_self">@</a> <font color="Maroon">nt</font></span>)</span>,<font color="Maroon">U1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="lang1.html#K15">*</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E21:32_1"/><i><font color="Green">E68</font></i>: 
 <a href="freealg.html#K11" target="_self">@</a> <font color="Maroon">nt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="freealg.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E22:32_1"/>
(  <a href="finseq_1.html#K3">len</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">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p2">(<span class="default"><a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</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">f</font>,<font color="Maroon">D</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> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="unialg_2.html#T3">UNIALG_2:3</a></i>;<br/>
<a NAME="E23:32_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="freealg.html#K1" target="_self">oper</a> <span class="p1">(<span class="default"><a href="freealg.html#K11" target="_self">@</a> <font color="Maroon">nt</font></span>)</span>,<font color="Maroon">U1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, , <a class="txt" href="freealg.html#E1:32"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<i><font color="Green">E72</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o2</font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> </span>}</span> 

;<br/>
<a NAME="E25:32_1"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="lang1.html#K15">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/>
<a NAME="E26:32_1"/>
 <a href="unialg_1.html#K4">signature</a> <span class="p1">(<span class="default"><a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<a NAME="E27:32_1"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , , , <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">ff</font> <a href="lang1.html#K15">*</a> <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  ;<br/>
<a NAME="E29:32_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">fx</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E30:32_1"/><i><font color="Green">E111</font></i>: 
 <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="hidden.html#R1">=</a> <font color="Maroon">nt</font>
 
<b>by </b><i>, , </i>;<br/>
<i><font color="Green">E112</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">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> = 
<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>, , </i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</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">g</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>reconsider </b><font color="Maroon">xa</font> = <font color="Maroon">xa</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="E33:32_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">xa</font> <a href="hidden.html#R2">in</a>  <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">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="freealg.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E34:32_1"/>
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a>  <a href="freealg.html#K5" target="_self">FreeOpNSG</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E35:32_1"/><b>then </b>
<font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">nt</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">xa</font>
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E36:32_1"/>
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="lang1.html#K15">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="freealg.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="freealg.html#D2" target="_self">Def2</a>, <a class="ref" href="freealg.html#D4" target="_self">Def4</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:32"/>
<font color="Maroon">ff</font> <a href="alg_1.html#R1">is_homomorphism</a>  <a href="freealg.html#K7" target="_self">FreeUnivAlgNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>,<font color="Maroon">U1</font>
 <b>by </b><i>, <a class="ref" href="alg_1.html#D1">ALG_1:def 1</a></i>;<br/>

<a NAME="E8:32"/><i><font color="Green">E116</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="freealg.html#K9" target="_self">FreeGenSetNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">ff</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="freealg.html#K9" target="_self">FreeGenSetNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</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">f</font>,<font color="Maroon">D</font></span>)</span> &amp; 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">f</font>,<font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="freealg.html#K9" target="_self">FreeGenSetNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="freealg.html#K9" target="_self">FreeGenSetNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> &amp;  <a href="freealg.html#K9" target="_self">FreeGenSetNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E117</font></i>: 
<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="E2:32_2"/><b>then </b>
<font color="Maroon">x</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">t</font></span>)</span> where <font color="Olive">t</font> is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> : <font color="Olive">t</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">f</font>,<font color="Maroon">D</font></span>)</span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E4:32_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</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">f</font>,<font color="Maroon">D</font></span>)</span> )
 ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="freealg.html#K9" target="_self">FreeGenSetNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="freealg.html#K10" target="_self">pi</a> <font color="Maroon">F</font>,<font color="Maroon">s</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:32"/>
<font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="freealg.html#K9" target="_self">FreeGenSetNSG</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
