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

<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">S</font>;<br/>



<b>set </b><font color="Maroon" title="c3">FA</font> =  <a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c4">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="c5">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="c6">FG</font> =  <a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>let </b><font color="Maroon" title="c7">U1</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osafree.html#D2" target="_self" title="OSAFREE:def.2">OSAFREE:def 2</a><br/><b>let </b><font color="Maroon" title="c8">F</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of  <a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</a> <font color="Maroon" title="c2">X</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">U1</font>;<br/>



<b>set </b><font color="Maroon" title="c9">SA</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#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">S1</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">U1</font>;<br/>
<b>set </b><font color="Maroon" title="c11">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="c12">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>reconsider </b><font color="Maroon" title="c13">NH1</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> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</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>,<span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c14">NHP</font> = <font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<div><i><font color="Green" title="E56">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><a NAME="E1:82_1"/><i><font color="Green" title="E57">A2</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="ref" href="osafree.html#D25" target="_self" title="OSAFREE:def.25">Def25</a></i>;<br/><a NAME="E2:82_1"/><i><font color="Green" title="E58">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p2">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="msafree.html#D1" title="MSAFREE:def.1">MSAFREE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:82_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p2">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="osafree.html#D25" target="_self" title="OSAFREE:def.25">Def25</a></i>;<br/><b>thus </b><a NAME="E4:82_1"/>
 <a href="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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="c15">s</font>
 ;<br/><b>thus </b><a NAME="E5:82_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c15">s</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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#K16" title="OSAFREE:func.16">FreeOSA</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="c15">s</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>thus </b><a NAME="E6:82_1"/><i><font color="Green" title="E59">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:82_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:82_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c15">s</font> iff  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:82_1_1"/>
( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c15">s</font> implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c15">s</font>
 ;<br/>

<a NAME="E2:82_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">y</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="c15">s</font>,<font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D27" target="_self" title="OSAFREE:def.27">Def27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:82_1_1_1"/><i><font color="Green" title="E60">A5</font></i>: 
( <font color="Maroon" title="c17">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="c15">s</font> &amp; <font color="Maroon" title="c16">y</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="c15">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#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c15">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="osafree.html#D26" target="_self" title="OSAFREE:def.26">Def26</a></i>;<br/>
<b>take </b><font color="Maroon" title="c18">x</font> =  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c15">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>

<a NAME="E5:82_1_1_1"/>
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c15">s</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E6:82_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="osafree.html#E4:82_1_1_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="osafree.html#D24" target="_self" title="OSAFREE:def.24">Def24</a></i>;<br/>
<b>hence </b><a NAME="E6:82_1_1_1"/>
( <font color="Maroon" title="c18">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E2:82_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="osafree.html#E4:82_1_1_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:82_1_1"/><i><font color="Green" title="E60">A6</font></i>: 
( <font color="Maroon" title="c17">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c18">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:82_1_1"/><i><font color="Green" title="E61">A7</font></i>: 
( <font color="Maroon" title="c18">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="c15">s</font> &amp; <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c15">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="osafree.html#E1:82_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="osafree.html#E3:82_1_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="osafree.html#D24" target="_self" title="OSAFREE:def.24">Def24</a></i>;<br/>
<a NAME="E6:82_1_1"/>
<font color="Maroon" title="c16">y</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="c15">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#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c15">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E2:82_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="osafree.html#E3:82_1_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="osafree.html#E5:82_1_1"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:82_1_1"/><b>then </b>
<font color="Maroon" title="c16">y</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="c15">s</font>,<font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E5:82_1_1"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="osafree.html#D26" target="_self" title="OSAFREE:def.26">Def26</a></i>;<br/>
<b>hence </b><a NAME="E8:82_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c15">s</font>
 <b>by </b><i><a class="ref" href="osafree.html#D27" target="_self" title="OSAFREE:def.27">Def27</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:82_1"/><i><font color="Green" title="E60">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c15">s</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:82_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p2">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c15">s</font>
 <b>by </b><i><a class="txt" href="osafree.html#E6:82_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/><b>thus </b><a NAME="E10:82_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p2">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c15">s</font>,<span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c15">s</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:82_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="osafree.html#E6:82_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="osafree.html#E8:82_1"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:82"/><b>then </b>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p2">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">NHP</font> = <font color="Maroon" title="c13">NH1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</a> <font color="Maroon" title="c2">X</font></span>)</span> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of  <a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</a> <font color="Maroon" title="c2">X</font>, <a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</a> <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">h</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</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>,<font color="Maroon" title="c7">U1</font><b> such that </b><br/><a NAME="E6:82"/><i><font color="Green" title="E57">A9</font></i>: 
( <font color="Maroon" title="c16">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c7">U1</font> &amp; <font color="Maroon" title="c16">h</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> &amp; <font color="Maroon" title="c16">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c15">NHP</font> )
 <b>by </b><i><a class="ref" href="osafree.html#T38" target="_self" title="OSAFREE:th.38">Th38</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">hCng</font> =  <a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c16">h</font> as  <a href="osalg_4.html#V3" title="OSALG_4:attr.3">monotone</a> <a href="osalg_4.html#NM4" title="OSALG_4:NM.4">OSCongruence</a> of  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="osafree.html#E6:82"><i><font color="Green" title="E57">A9</font></i></a>, <a class="ref" href="osalg_4.html#T26" title="OSALG_4:th.26">OSALG_4:26</a></i>;<br/>
<a NAME="E8:82"/><i><font color="Green" title="E58">A10</font></i>: 
 <a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c17">hCng</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D18" target="_self" title="OSAFREE:def.18">Def18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">H</font> =  <a href="osalg_4.html#K23" title="OSALG_4:func.23">OSHomQuot</a> <font color="Maroon" title="c16">h</font>,<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> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c7">U1</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c18">H</font>
;<br/>

<b>thus </b><a NAME="E10:82"/>
( <font color="Maroon" title="c18">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c7">U1</font> &amp; <font color="Maroon" title="c18">H</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> )
 <b>by </b><i><a class="txt" href="osafree.html#E6:82"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="osafree.html#E8:82"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="osalg_4.html#T27" title="OSALG_4:th.27">OSALG_4:27</a></i>;<br/>

<a NAME="E11:82"/>
for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>  st <font color="Olive" title="b2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Maroon" title="c8">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c20">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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#K16" title="OSAFREE:func.16">FreeOSA</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="c19">s</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>;<br/>



<b>assume </b><a NAME="E1:82_2"/><i><font color="Green" title="E59">A11</font></i>: 
<font color="Maroon" title="c20">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font>
 ;<br/>

<a NAME="E2:82_2"/><i><font color="Green" title="E60">A12</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</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="c19">s</font>,<font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D27" target="_self" title="OSAFREE:def.27">Def27</a></i>;<br/>
<a NAME="E3:82_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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#K16" title="OSAFREE:func.16">FreeOSA</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="c19">s</font>
 
;<br/>
<a NAME="E4:82_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K22" title="OSALG_4:func.22">OSHomQuot</a> <font color="Maroon" title="c16">h</font>,<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="c19">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:82_2"/><b>then </b><i><font color="Green" title="E61">A13</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K22" title="OSALG_4:func.22">OSHomQuot</a> <font color="Maroon" title="c16">h</font>,<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>,<font color="Maroon" title="c19">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K22" title="OSALG_4:func.22">OSHomQuot</a> <font color="Maroon" title="c16">h</font>,<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>,<font color="Maroon" title="c19">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K22" title="OSALG_4:func.22">OSHomQuot</a> <font color="Maroon" title="c16">h</font>,<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="c19">s</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E6:82_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K22" title="OSALG_4:func.22">OSHomQuot</a> <font color="Maroon" title="c16">h</font>,<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>,<font color="Maroon" title="c19">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">OSF</font> = <span class="p1">(<span class="default"><a href="osalg_4.html#K22" title="OSALG_4:func.22">OSHomQuot</a> <font color="Maroon" title="c16">h</font>,<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="c19">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> <b>by </b><i><a class="txt" href="osafree.html#E5:82_2"><i><font color="Green" title="E61">A13</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:82_2_1"/><i><font color="Green" title="E62">A14</font></i>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</font>
 ;<br/><b>consider </b><font color="Maroon" title="c23">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:82_2_1"/><i><font color="Green" title="E63">A15</font></i>: 
( <font color="Maroon" title="c23">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="c19">s</font> &amp; <font color="Maroon" title="c22">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="c19">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#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c19">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="osafree.html#E2:82_2"><i><font color="Green" title="E60">A12</font></i></a>, <a class="txt" href="osafree.html#E1:82_2_1"><i><font color="Green" title="E62">A14</font></i></a>, <a class="ref" href="osafree.html#D26" target="_self" title="OSAFREE:def.26">Def26</a></i>;<br/><a NAME="E4:82_2_1"/><i><font color="Green" title="E64">A16</font></i>: 
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c19">s</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="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="osafree.html#E3:82_2_1"><i><font color="Green" title="E63">A15</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="c24">xa</font> =  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c19">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> 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="c19">s</font> <b>by </b><i><a class="txt" href="osafree.html#E3:82_2_1"><i><font color="Green" title="E63">A15</font></i></a>, <a class="ref" href="osafree.html#T10" target="_self" title="OSAFREE:th.10">Th10</a></i>;<br/><a NAME="E6:82_2_1"/><i><font color="Green" title="E65">A17</font></i>: 
<font color="Maroon" title="c24">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c19">s</font>
 <b>by </b><i><a class="txt" href="osafree.html#E4:82_2_1"><i><font color="Green" title="E64">A16</font></i></a>, <a class="ref" href="osafree.html#D25" target="_self" title="OSAFREE:def.25">Def25</a></i>;<br/><i><font color="Green" title="E66">A18</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="c24">xa</font> = 
<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="c19">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">xa</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/>.= 
<font color="Maroon" title="c22">x</font>
<b>by </b><i><a class="txt" href="osafree.html#E3:82_2_1"><i><font color="Green" title="E63">A15</font></i></a>, <a class="ref" href="osalg_4.html#D24" title="OSALG_4:def.24">OSALG_4:def 24</a></i>
;<br/><i><font color="Green" title="E67">A19</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">NHP</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">xa</font> = 
<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="c19">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">xa</font>
<b>by </b><i><a class="txt" href="osafree.html#E2:82"><i><font color="Green" title="E56">A1</font></i></a></i>
<br/>.= 
<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="c19">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">xa</font>
<b>by </b><i><a class="txt" href="osafree.html#E4:82_2_1"><i><font color="Green" title="E64">A16</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><a NAME="E9:82_2_1"/><i><font color="Green" title="E68">A20</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K22" title="OSALG_4:func.22">OSHomQuot</a> <font color="Maroon" title="c16">h</font>,<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="c19">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <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="c24">xa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">xa</font>
 <b>by </b><i><a class="txt" href="osafree.html#E6:82"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="osafree.html#E8:82"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="osalg_4.html#D29" title="OSALG_4:def.29">OSALG_4:def 29</a></i>;<br/><i><font color="Green" title="E69">A21</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c19">s</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c15">NHP</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font>
<b>by </b><i><a class="txt" href="osafree.html#E6:82"><i><font color="Green" title="E57">A9</font></i></a>, <a class="ref" href="msafree.html#D1" title="MSAFREE:def.1">MSAFREE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">NHP</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
;<br/><a NAME="E11:82_2_1"/><i><font color="Green" title="E70">A22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K22" title="OSAFREE:func.22">PTVars</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="c19">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">xa</font>
 <b>by </b><i><a class="txt" href="osafree.html#E6:82_2_1"><i><font color="Green" title="E65">A17</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E12:82_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">NHP</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K21" title="OSAFREE:func.21">PTVars</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="osafree.html#E2:82"><i><font color="Green" title="E56">A1</font></i></a></i>;<br/><a NAME="E13:82_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font>
 <b>by </b><i><a class="txt" href="osafree.html#E3:82_2_1"><i><font color="Green" title="E63">A15</font></i></a>, <a class="txt" href="osafree.html#E4:82_2_1"><i><font color="Green" title="E64">A16</font></i></a>, <a class="txt" href="osafree.html#E8:82_2_1"><i><font color="Green" title="E67">A19</font></i></a>, <a class="txt" href="osafree.html#E10:82_2_1"><i><font color="Green" title="E69">A21</font></i></a>, <a class="txt" href="osafree.html#E11:82_2_1"><i><font color="Green" title="E70">A22</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E14:82_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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#K22" title="OSALG_4:func.22">OSHomQuot</a> <font color="Maroon" title="c16">h</font>,<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>,<font color="Maroon" title="c19">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:82_2_1"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="osafree.html#E7:82_2_1"><i><font color="Green" title="E66">A18</font></i></a>, <a class="txt" href="osafree.html#E9:82_2_1"><i><font color="Green" title="E68">A20</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:82_2"/><b>then </b>
<font color="Maroon" title="c8">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c21">OSF</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>
<b>hence </b><a NAME="E10:82_2"/>
<font color="Maroon" title="c8">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</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="c19">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E1:82_2"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="osalg_4.html#D30" title="OSALG_4:def.30">OSALG_4:def 30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:82"/><b>then </b>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E13:82"/>
<font color="Maroon" title="c18">H</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" title="OSAFREE:func.24">OSFreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">F</font>
 <b>by </b><i><a class="ref" href="msafree.html#D1" title="MSAFREE:def.1">MSAFREE:def 1</a></i>;<br/>


</div>
