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

<b>let </b><font color="Maroon" title="c1">f</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="relat_1.html#V2" title="RELAT_1:attr.2">with_zero</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">D</font> be  <a href="freealg.html#V1" title="FREEALG:attr.1">disjoint_with_NAT</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">fgs</font> =  <a href="freealg.html#K17" title="FREEALG:func.17">FreeGenSetZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>;<br/>
<b>set </b><font color="Maroon" title="c4">fua</font> =  <a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>;<br/>
<b>let </b><font color="Maroon" title="c5">U1</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="freealg.html#D6" target="_self" title="FREEALG:def.6">FREEALG:def 6</a><br/>

<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E27">A1</font></i>: 
 <a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">U1</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c6">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="freealg.html#K17" title="FREEALG:func.17">FreeGenSetZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font>;<br/>

<b>set </b><font color="Maroon" title="c7">A</font> =  <a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>;<br/>
<b>set </b><font color="Maroon" title="c8">c1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font>;<br/>
<b>set </b><font color="Maroon" title="c9">cf</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font> =  <a href="freealg.html#K18" title="FREEALG:func.18">pi</a> <font color="Maroon" title="c6">F</font>,$1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>,  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font> = <span class="p1">(<span class="default"><a href="freealg.html#K1" title="FREEALG:func.1">oper</a> <span class="p2">(<span class="default"><a href="freealg.html#K11" title="FREEALG:func.11">@</a> $1</span>)</span>,<font color="Maroon" title="c5">U1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $3;<br/>
<b>consider </b><font color="Maroon" title="c10">ff</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font><b> such that </b><br/><a NAME="E3:46"/><i><font color="Green" title="E28">A2</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> holds <br/><font color="Maroon" title="c10">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">t</font>)
 <b>and </b><br/><a NAME="E4:46"/><i><font color="Green" title="E29">A3</font></i>: 
for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span><br/> for <font color="Olive" title="b2">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> holds <br/><font color="Maroon" title="c10">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font>,<font color="Maroon" title="c10">ff</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b2">ts</font>)
 <b>from </b><i><a class="ref" href="dtconstr.html#S8" title="DTCONSTR:sch.8">DTCONSTR:sch 8</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c11">ff</font> = <font color="Maroon" title="c10">ff</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>,<font color="Maroon" title="c5">U1</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c11">ff</font>
;<br/>

<a NAME="E6:46"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> holds <br/>for <font color="Olive" title="b2">o1</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span><br/> for <font color="Olive" title="b3">o2</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c5">U1</font>  st <font color="Olive" title="b2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c5">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/>for <font color="Olive" title="b4">x</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">o1</font> holds <br/><font color="Maroon" title="c11">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ff</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b4">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:46_1"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c13">o1</font> be   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c14">o2</font> be   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c5">U1</font>;<br/>



<b>assume </b><a NAME="E2:46_1"/><i><font color="Green" title="E31">A5</font></i>: 
( <font color="Maroon" title="c13">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c14">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c5">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c15">x</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>;<br/>

<b>assume </b><a NAME="E3:46_1"/><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">o1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c16">xa</font> = <font color="Maroon" title="c15">x</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> ;<br/>
<a NAME="E5:46_1"/><i><font color="Green" title="E33">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="freealg.html#K14" title="FREEALG:func.14">FreeOpSeqZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="freealg.html#K14" title="FREEALG:func.14">FreeOpSeqZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="freealg.html#K14" title="FREEALG:func.14">FreeOpSeqZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="freealg.html#D18" target="_self" title="FREEALG:def.18">Def18</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:46_1"/><b>then </b><i><font color="Green" title="E34">A8</font></i>: 
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="freealg.html#E1:46_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="freealg.html#T7" target="_self" title="FREEALG:th.7">Th7</a></i>;<br/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">o1</font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
<b>by </b><i><a class="ref" href="unialg_2.html#T2" title="UNIALG_2:th.2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o1</font> </span>}</span> 

;<br/>
<b>then consider </b><font color="Maroon" title="c17">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E9:46_1"/><i><font color="Green" title="E35">A9</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">w</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o1</font> )
 <b>by </b><i><a class="txt" href="freealg.html#E3:46_1"><i><font color="Green" title="E32">A6</font></i></a></i>;<br/>
<a NAME="E10:46_1"/><i><font color="Green" title="E36">A10</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p2">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o1</font> )
 
<b>by </b><i><a class="txt" href="freealg.html#E2:46_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="freealg.html#E6:46_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c16">xa</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">xa</font>
<b>by </b><i><a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">xa</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E12:46_1"/><b>then </b><i><font color="Green" title="E37">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c16">xa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">xa</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">nt</font> = <font color="Maroon" title="c12">n</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <b>by </b><i><a class="txt" href="freealg.html#E1:46_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="freealg.html#E5:46_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">nd</font> = <font color="Maroon" title="c12">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="txt" href="freealg.html#E1:46_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="freealg.html#E5:46_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">xa</font> = <font color="Maroon" title="c16">xa</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c21">rxa</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c20">xa</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">D</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c22">rxa</font> = <font color="Maroon" title="c21">rxa</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E18:46_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">nd</font>,<font color="Maroon" title="c22">rxa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="freealg.html#K2" title="FREEALG:func.2">REL</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:46_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="freealg.html#E5:46_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="freealg.html#E9:46_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="freealg.html#E10:46_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="freealg.html#E12:46_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="freealg.html#D8" target="_self" title="FREEALG:def.8">Def8</a></i>;<br/>
<a NAME="E19:46_1"/><b>then </b><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c18">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c20">xa</font>
 
<b>by </b><i><a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/>
<a NAME="E20:46_1"/><b>then </b><i><font color="Green" title="E39">A13</font></i>: 
<font color="Maroon" title="c11">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c20">xa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="freealg.html#K1" title="FREEALG:func.1">oper</a> <span class="p2">(<span class="default"><a href="freealg.html#K11" title="FREEALG:func.11">@</a> <font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c5">U1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ff</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c15">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="freealg.html#E4:46"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/>
<a NAME="E21:46_1"/><i><font color="Green" title="E40">A14</font></i>: 
 <a href="freealg.html#K11" title="FREEALG:func.11">@</a> <font color="Maroon" title="c18">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E19:46_1"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="freealg.html#D16" target="_self" title="FREEALG:def.16">Def16</a></i>;<br/>
<a NAME="E22:46_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c5">U1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p2">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c5">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c5">U1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="freealg.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="unialg_2.html#T3" title="UNIALG_2:th.3">UNIALG_2:3</a></i>;<br/>
<a NAME="E23:46_1"/><b>then </b><i><font color="Green" title="E41">A15</font></i>: 
 <a href="freealg.html#K1" title="FREEALG:func.1">oper</a> <span class="p1">(<span class="default"><a href="freealg.html#K11" title="FREEALG:func.11">@</a> <font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c5">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">o2</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:46_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="freealg.html#E2:46_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="freealg.html#E21:46_1"><i><font color="Green" title="E40">A14</font></i></a>, <a class="ref" href="freealg.html#D4" target="_self" title="FREEALG:def.4">Def4</a></i>;<br/>
<i><font color="Green" title="E42">A16</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">o2</font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c14">o2</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T2" title="UNIALG_2:th.2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c14">o2</font> </span>}</span> 

;<br/>
<a NAME="E25:46_1"/><i><font color="Green" title="E43">A17</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ff</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">x</font>
 
<b>by </b><i><a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a></i>;<br/>
<a NAME="E26:46_1"/>
 <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c5">U1</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="unialg_2.html#D2" title="UNIALG_2:def.2">UNIALG_2:def 2</a></i>;<br/>
<a NAME="E27:46_1"/><b>then </b><i><font color="Green" title="E44">A18</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c14">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">x</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:46_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="freealg.html#E2:46_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="freealg.html#E5:46_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="freealg.html#E9:46_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="freealg.html#E10:46_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">fx</font> = <font color="Maroon" title="c11">ff</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c15">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  ;<br/>
<a NAME="E29:46_1"/><i><font color="Green" title="E45">A19</font></i>: 
<font color="Maroon" title="c23">fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c14">o2</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="freealg.html#E25:46_1"><i><font color="Green" title="E43">A17</font></i></a>, <a class="txt" href="freealg.html#E27:46_1"><i><font color="Green" title="E44">A18</font></i></a></i>;<br/>
<a NAME="E30:46_1"/><i><font color="Green" title="E46">A20</font></i>: 
 <a href="freealg.html#K4" title="FREEALG:func.4">Sym</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">nt</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:46_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="freealg.html#E5:46_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="freealg.html#D10" target="_self" title="FREEALG:def.10">Def10</a></i>;<br/>
<i><font color="Green" title="E47">A21</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="freealg.html#K13" title="FREEALG:func.13">FreeOpZAO</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="freealg.html#E1:46_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="freealg.html#E5:46_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="freealg.html#D17" target="_self" title="FREEALG:def.17">Def17</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> </span>}</span> 

;<br/>
<b>reconsider </b><font color="Maroon" title="c24">xa</font> = <font color="Maroon" title="c20">xa</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E33:46_1"/><i><font color="Green" title="E48">A22</font></i>: 
<font color="Maroon" title="c24">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="freealg.html#K13" title="FREEALG:func.13">FreeOpZAO</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="freealg.html#E9:46_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="freealg.html#E10:46_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="freealg.html#E31:46_1"><i><font color="Green" title="E47">A21</font></i></a></i>;<br/>
<a NAME="E34:46_1"/>
<font color="Maroon" title="c13">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="freealg.html#K13" title="FREEALG:func.13">FreeOpZAO</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:46_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="freealg.html#E2:46_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="freealg.html#D18" target="_self" title="FREEALG:def.18">Def18</a></i>;<br/>
<a NAME="E35:46_1"/><b>then </b>
<font color="Maroon" title="c13">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c24">xa</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:46_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="freealg.html#E5:46_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="freealg.html#E30:46_1"><i><font color="Green" title="E46">A20</font></i></a>, <a class="txt" href="freealg.html#E33:46_1"><i><font color="Green" title="E48">A22</font></i></a>, <a class="ref" href="freealg.html#D17" target="_self" title="FREEALG:def.17">Def17</a></i>;<br/>
<b>hence </b><a NAME="E36:46_1"/>
<font color="Maroon" title="c11">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ff</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="freealg.html#E20:46_1"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="freealg.html#E23:46_1"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="freealg.html#E24:46_1"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="freealg.html#E29:46_1"><i><font color="Green" title="E45">A19</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:46"/>
<font color="Maroon" title="c11">ff</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a>  <a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">U1</font>
 <b>by </b><i><a class="txt" href="freealg.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="alg_1.html#D1" title="ALG_1:def.1">ALG_1:def 1</a></i>;<br/>

<a NAME="E8:46"/><i><font color="Green" title="E30">A23</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ff</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="freealg.html#K17" title="FREEALG:func.17">FreeGenSetZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">ff</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="freealg.html#K17" title="FREEALG:func.17">FreeGenSetZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="freealg.html#K17" title="FREEALG:func.17">FreeGenSetZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="freealg.html#K17" title="FREEALG:func.17">FreeGenSetZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> &amp;  <a href="freealg.html#K17" title="FREEALG:func.17">FreeGenSetZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:46_2"/><i><font color="Green" title="E31">A24</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 ;<br/><a NAME="E2:46_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c13">s</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span><b> such that </b><br/><a NAME="E4:46_2"/><i><font color="Green" title="E32">A25</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c13">s</font> &amp; <font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> )
 ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c11">ff</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="freealg.html#K17" title="FREEALG:func.17">FreeGenSetZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> = 
<font color="Maroon" title="c11">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="freealg.html#E8:46"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="freealg.html#E1:46_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="freealg.html#K18" title="FREEALG:func.18">pi</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c13">s</font>
<b>by </b><i><a class="txt" href="freealg.html#E3:46"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="freealg.html#E4:46_2"><i><font color="Green" title="E32">A25</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="freealg.html#E4:46_2"><i><font color="Green" title="E32">A25</font></i></a>, <a class="ref" href="freealg.html#D21" target="_self" title="FREEALG:def.21">Def21</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:46"/>
<font color="Maroon" title="c11">ff</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="freealg.html#K17" title="FREEALG:func.17">FreeGenSetZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="freealg.html#E8:46"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
