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<b>let </b><font color="Maroon">S</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">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>set </b><font color="Maroon">PTA</font> =  <a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">P</font> =  <a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font>;<br/>
<b>thus </b><a NAME="E1:59"/>
 <a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font> is <a href="osalg_4.html#V3">monotone</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">o2</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_4.html#D28">OSALG_4:def 28</a><br/>

<b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">o1</font> <a href="osalg_1.html#R6">&lt;=</a> <font color="Maroon">o2</font>
 ;<br/>

<b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>;<br/>



<b>assume </b><a NAME="E2:59_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span>
 ;<br/>

<a NAME="E3:59_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">o1</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o2</font> &amp;  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font> <a href="osalg_1.html#R2">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o2</font> )
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#D22">OSALG_1:def 22</a></i>;<br/>
<a NAME="E4:59_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_1.html#D7">OSALG_1:def 7</a></i>;<br/>
<a NAME="E5:59_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:59_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<b>hence </b><a NAME="E7:59_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o2</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/>


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


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