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<b>let </b><font color="Maroon">S</font> be  <a href="osalg_1.html#V8">monotone</a> <a href="osalg_1.html#V10">regular</a> <a href="osalg_4.html#V2">locally_directed</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="relat_1.html#V2">V5</a> <a href="struct_0.html#NM8">ManySortedSet</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">t1</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">X</font></span>)</span>, <font color="Maroon">t2</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">X</font></span>)</span>;<br/>





<b>set </b><font color="Maroon">R</font> =  <a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font>;<br/>
<b>thus </b><a NAME="E1:88"/>
( <font color="Maroon">t2</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#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">t1</font> implies <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E2:88"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font>
 ;<br/>

<a NAME="E3:88"/>
( <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</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#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">t2</font> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</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#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">t1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:88"/><b>then </b>
(  <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">X</font></span>)</span>,<font color="Maroon">t1</font> <a href="hidden.html#R1">=</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">X</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span> &amp;  <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">X</font></span>)</span>,<font color="Maroon">t2</font> <a href="hidden.html#R1">=</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">X</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:88"/>
<font color="Maroon">t2</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#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">t1</font>
 <b>by </b><i>, </i>;<br/>


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