<?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>let </b><font color="Maroon">c<sub>3</sub></font> be   <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>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</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#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</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>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> 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> <span class="p1">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="osafree.html#D12" target="_self">Def12</a></i>;<br/>
<i><font color="Green">E51</font></i>:  <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="osalg_4.html#K8">OSClass</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self">Def28</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5">CompClass</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3">CComp</a> <span class="p3">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>8</sub></font>

;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="osalg_4.html#K3">CComp</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="osalg_4.html#K5">CompClass</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3">CComp</a> <span class="p2">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:75_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:75_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K5">CompClass</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3">CComp</a> <span class="p2">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E2:75"><i><font color="Green">E51</font></i></a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:75_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K3">CComp</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="osalg_4.html#D11">OSALG_4:def 11</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></font> 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>11</sub></font> <b>by </b><i><a class="txt" href="osafree.html#E4:75_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b> <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="osalg_4.html#K8">OSClass</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self">Def28</a></i>
<br/>.= 
 <a href="osalg_4.html#K8">OSClass</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="osafree.html#E4:75_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="osalg_4.html#T13">OSALG_4:13</a></i>
<br/>.= 
 <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self">Def28</a></i>
;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:75"/>
 <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>hence </b><a NAME="E5:75"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="osafree.html#T33" target="_self">Th33</a></i>;<br/>


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