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

<b>set </b><font color="Maroon" title="c4">NH</font> =  <a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">A</font> =  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c6">R</font> =  <a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c7">PTA</font> =  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c8">SPTA</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">NHs</font> =  <a href="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">s</font>;<br/>
<b>set </b><font color="Maroon" title="c10">D</font> =  <a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<a NAME="E1:80_1_1"/><i><font color="Green" title="E55">A1</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D24" title="OSALG_4:def.24">OSALG_4:def 24</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>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="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <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> holds <br/>$2 <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="Olive" title="b1">t</font>;<br/>
<a NAME="E2:80_1_1"/><i><font color="Green" title="E56">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font> holds <br/> ex <font color="Olive" title="b2">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">a</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:80_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E2:80_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p5">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p5">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<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></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="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : ( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">t</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="osafree.html#T31" target="_self" title="OSAFREE:th.31">Th31</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">t</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E4:80_1_1_1"/><i><font color="Green" title="E57">A3</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <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="Maroon" title="c12">t</font></span>)</span> &amp; <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c12">t</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c13">s1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c14">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:80_1_1_1"/><i><font color="Green" title="E58">A4</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s1</font> &amp; <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c13">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="osafree.html#E4:80_1_1_1"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="osafree.html#T4" target="_self" title="OSAFREE:th.4">Th4</a></i>;<br/>
<a NAME="E7:80_1_1_1"/><i><font color="Green" title="E59">A5</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s1</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E4:80_1_1_1"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="osafree.html#E6:80_1_1_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>take </b>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c12">t</font>
;<br/>

<b>thus </b><a NAME="E8:80_1_1_1"/>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="osafree.html#E6:80_1_1_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="osafree.html#E7:80_1_1_1"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="osafree.html#D24" target="_self" title="OSAFREE:def.24">Def24</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c15">rt</font> =  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c12">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> <b>by </b><i><a class="txt" href="osafree.html#E6:80_1_1_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="osafree.html#E7:80_1_1_1"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="osafree.html#T10" target="_self" title="OSAFREE:th.10">Th10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">tt</font> =  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c12">t</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E6:80_1_1_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="osafree.html#T10" target="_self" title="OSAFREE:th.10">Th10</a></i>;<br/>
<a NAME="E11:80_1_1_1"/><i><font color="Green" title="E60">A6</font></i>: 
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c16">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c15">rt</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></i>;<br/>
<a NAME="E12:80_1_1_1"/><i><font color="Green" title="E61">A7</font></i>: 
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c16">tt</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="osafree.html#T33" target="_self" title="OSAFREE:th.33">Th33</a></i>;<br/>
<a NAME="E13:80_1_1_1"/><i><font color="Green" title="E62">A8</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">s</font></span>)</span> <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="Maroon" title="c12">t</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E4:80_1_1_1"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="osalg_4.html#D24" title="OSALG_4:def.24">OSALG_4:def 24</a></i>;<br/>
<a NAME="E14:80_1_1_1"/><i><font color="Green" title="E63">A9</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">rt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c15">rt</font>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D23" title="OSALG_4:def.23">OSALG_4:def 23</a></i>;<br/>
<a NAME="E15:80_1_1_1"/><i><font color="Green" title="E64">A10</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="osafree.html#E11:80_1_1_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="osafree.html#E12:80_1_1_1"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="osafree.html#E13:80_1_1_1"><i><font color="Green" title="E62">A8</font></i></a>, <a class="ref" href="osalg_4.html#D23" title="OSALG_4:def.23">OSALG_4:def 23</a></i>;<br/>
<b>let </b><font color="Maroon" title="c17">t1</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E16:80_1_1_1"/><i><font color="Green" title="E65">A11</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <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="Maroon" title="c17">t1</font></span>)</span>
 ;<br/>

<a NAME="E17:80_1_1_1"/><b>then </b><i><font color="Green" title="E66">A12</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">s</font></span>)</span> <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="Maroon" title="c17">t1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D24" title="OSALG_4:def.24">OSALG_4:def 24</a></i>;<br/>
<a NAME="E18:80_1_1_1"/>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c17">t1</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="osalg_4.html#K17" title="OSALG_4:func.17">OSNat_Hom</a> <span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E1:80_1_1"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="osafree.html#E15:80_1_1_1"><i><font color="Green" title="E64">A10</font></i></a>, <a class="txt" href="osafree.html#E16:80_1_1_1"><i><font color="Green" title="E65">A11</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">rt1</font> =  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c17">t1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c19">tt2</font> = <font color="Maroon" title="c18">rt1</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="ref" href="osafree.html#T16" target="_self" title="OSAFREE:th.16">Th16</a></i>;<br/>
<a NAME="E21:80_1_1_1"/><i><font color="Green" title="E67">A13</font></i>: 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c18">rt1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c15">rt</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E13:80_1_1_1"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="osafree.html#E14:80_1_1_1"><i><font color="Green" title="E63">A9</font></i></a>, <a class="txt" href="osafree.html#E17:80_1_1_1"><i><font color="Green" title="E66">A12</font></i></a>, <a class="ref" href="osalg_4.html#D23" title="OSALG_4:def.23">OSALG_4:def 23</a></i>;<br/>
<a NAME="E22:80_1_1_1"/>
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c19">tt2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c18">rt1</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></i>;<br/>
<a NAME="E23:80_1_1_1"/><b>then </b>
<font color="Maroon" title="c19">tt2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c16">tt</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E11:80_1_1_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="osafree.html#E21:80_1_1_1"><i><font color="Green" title="E67">A13</font></i></a>, <a class="ref" href="osafree.html#T35" target="_self" title="OSAFREE:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E24:80_1_1_1"/>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c12">t</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="c17">t1</font>
 <b>by </b><i><a class="txt" href="osafree.html#E6:80_1_1_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="osafree.html#T37" target="_self" title="OSAFREE:th.37">Th37</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c11">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:80_1_1"/><i><font color="Green" title="E57">A14</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="osafree.html#E2:80_1_1"><i><font color="Green" title="E56">A2</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c11">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font>, <a href="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="osafree.html#E4:80_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c13">t</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E6:80_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <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="Maroon" title="c13">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font>
 ;<br/>

<b>hence </b><a NAME="E7:80_1_1"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p4">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c13">t</font></span>)</span></span>)</span> <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">t</font>
 <b>by </b><i><a class="txt" href="osafree.html#E4:80_1_1"><i><font color="Green" title="E57">A14</font></i></a></i>;<br/>


</div>
