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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="osalg_4.html#V2">locally_directed</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="relat_1.html#V2">V5</a> <a href="struct_0.html#NM9">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="osafree.html#K16" target="_self">FreeOSA</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#V13">monotone</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osafree.html#D2" target="_self">OSAFREE:def 2</a><br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" target="_self">FreeOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K16" target="_self">FreeOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<div><i><font color="Green">E56</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">c<sub>15</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:82_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K21" target="_self">PTVars</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="osafree.html#D25" target="_self">Def25</a></i>;<br/><a NAME="E2:82_1"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="msafree.html#D1">MSAFREE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:82_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="osafree.html#K21" target="_self">PTVars</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="osafree.html#D25" target="_self">Def25</a></i>;<br/><b>thus </b><a NAME="E4:82_1"/>
 <a href="osafree.html#K21" target="_self">PTVars</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>thus </b><a NAME="E5:82_1"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" target="_self">FreeOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>thus </b><a NAME="E6:82_1"/><i><font color="Green">E59</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K21" target="_self">PTVars</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:82_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:82_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> iff ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></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">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:82_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:82_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K23" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="osafree.html#D27" target="_self">Def27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:82_1_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="osafree.html#D26" target="_self">Def26</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>18</sub></font> =  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/>

<a NAME="E5:82_1_1_1"/>
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E6:82_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="osafree.html#E4:82_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="osafree.html#D24" target="_self">Def24</a></i>;<br/>
<b>hence </b><a NAME="E6:82_1_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="osafree.html#E2:82_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="osafree.html#E4:82_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:82_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:82_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="osafree.html#E1:82_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="osafree.html#E2:82_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="osafree.html#D24" target="_self">Def24</a></i>;<br/>
<a NAME="E5:82_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E2:82_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="osafree.html#E2:82_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="osafree.html#E4:82_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:82_1_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K23" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="osafree.html#E4:82_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="osafree.html#D26" target="_self">Def26</a></i>;<br/>
<b>hence </b><a NAME="E7:82_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="osafree.html#D27" target="_self">Def27</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:82_1"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:82_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="osafree.html#K21" target="_self">PTVars</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E6:82_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/><b>thus </b><a NAME="E10:82_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:82_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="osafree.html#E6:82_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="osafree.html#E8:82_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:82"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font>, <a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E6:82"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> is <a href="osalg_3.html#V1">order-sorted</a> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="osafree.html#T38" target="_self">Th38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> =  <a href="osalg_4.html#K19">OSCng</a> <font color="Maroon">c<sub>16</sub></font> as  <a href="osalg_4.html#V3">monotone</a> <a href="osalg_4.html#NM4">OSCongruence</a> of  <a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="osafree.html#E6:82"><i><font color="Green">E57</font></i></a>, <a class="ref" href="osalg_4.html#T26">OSALG_4:26</a></i>;<br/>
<a NAME="E8:82"/><i><font color="Green">E58</font></i>: 
 <a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="osafree.html#D18" target="_self">Def18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> =  <a href="osalg_4.html#K23">OSHomQuot</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" target="_self">FreeOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<b>thus </b><a NAME="E10:82"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="osafree.html#K16" target="_self">FreeOSA</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> is <a href="osalg_3.html#V1">order-sorted</a> )
 <b>by </b><i><a class="txt" href="osafree.html#E6:82"><i><font color="Green">E57</font></i></a>, <a class="txt" href="osafree.html#E8:82"><i><font color="Green">E58</font></i></a>, <a class="ref" href="osalg_4.html#T27">OSALG_4:27</a></i>;<br/>

<a NAME="E11:82"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" target="_self">FreeOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>19</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" target="_self">FreeOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>;<br/>



<b>assume </b><a NAME="E1:82_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<a NAME="E2:82_2"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K23" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="osafree.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E3:82_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" target="_self">FreeOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 
;<br/>
<a NAME="E4:82_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K22">OSHomQuot</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:82_2"/><b>then </b><i><font color="Green">E61</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K22">OSHomQuot</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p3">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K22">OSHomQuot</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p3">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K22">OSHomQuot</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E6:82_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K22">OSHomQuot</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p3">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><a href="osalg_4.html#K22">OSHomQuot</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <b>by </b><i><a class="txt" href="osafree.html#E5:82_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_2.html#T4">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">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:82_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:82_2_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="osafree.html#E2:82_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="osafree.html#E1:82_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="osafree.html#D26" target="_self">Def26</a></i>;<br/><a NAME="E4:82_2_1"/><i><font color="Green">E64</font></i>: 
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K21" target="_self">PTVars</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E3:82_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="osafree.html#D24" target="_self">Def24</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> =  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <b>by </b><i><a class="txt" href="osafree.html#E3:82_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="osafree.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E6:82_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E4:82_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="osafree.html#D25" target="_self">Def25</a></i>;<br/><i><font color="Green">E66</font></i>:  <a href="osalg_4.html#K8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K17">OSNat_Hom</a> <span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="ref" href="osalg_4.html#D23">OSALG_4:def 23</a></i>
<br/>.= 
<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="osafree.html#E3:82_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="osalg_4.html#D24">OSALG_4:def 24</a></i>
;<br/><i><font color="Green">E67</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></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">OSNat_Hom</a> <span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="osafree.html#K21" target="_self">PTVars</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="osafree.html#E2:82"><i><font color="Green">E56</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="osafree.html#E4:82_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E9:82_2_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K22">OSHomQuot</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8">OSClass</a> <span class="p2">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E6:82"><i><font color="Green">E57</font></i></a>, <a class="txt" href="osafree.html#E8:82"><i><font color="Green">E58</font></i></a>, <a class="ref" href="osalg_4.html#D29">OSALG_4:def 29</a></i>;<br/><i><font color="Green">E69</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="osafree.html#E6:82"><i><font color="Green">E57</font></i></a>, <a class="ref" href="msafree.html#D1">MSAFREE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
;<br/><a NAME="E11:82_2_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E6:82_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E12:82_2_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K21" target="_self">PTVars</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E2:82"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E13:82_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E3:82_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="osafree.html#E4:82_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="osafree.html#E8:82_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="osafree.html#E10:82_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="osafree.html#E11:82_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E14:82_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K22">OSHomQuot</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p3">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:82_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="osafree.html#E7:82_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="osafree.html#E9:82_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:82_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>
<b>hence </b><a NAME="E10:82_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E1:82_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="osalg_4.html#D30">OSALG_4:def 30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:82"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" target="_self">FreeOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E13:82"/>
<font color="Maroon">c<sub>18</sub></font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="osafree.html#K24" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="msafree.html#D1">MSAFREE:def 1</a></i>;<br/>


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