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<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="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</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>5</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>6</sub></font> =  <a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:77"/>
 <a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2">c=</a>  <a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="osafree.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E2:77"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="osafree.html#K25" target="_self">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>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="osafree.html#T36" target="_self">Th36</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:77"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>7</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>3</sub></font> &amp; <font color="Maroon">c<sub>4</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>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<b>thus </b><a NAME="E4:77"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">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>4</sub></font> implies <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="osafree.html#E2:77"><i><font color="Green">E53</font></i></a>, <a class="txt" href="osafree.html#E3:77"><i><font color="Green">E54</font></i></a>, <a class="ref" href="osafree.html#T34" target="_self">Th34</a></i>;<br/>

<b>assume </b><a NAME="E5:77"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>hence </b><a NAME="E6:77"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">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>4</sub></font>
 <b>by </b><i><a class="ref" href="osafree.html#T33" target="_self">Th33</a></i>;<br/>


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