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<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">R</font> be   <a href="osalg_4.html#NM4" title="OSALG_4:NM.4">OSCongruence</a> of  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>;<br/><b>let </b><font color="Maroon" title="c4">t1</font>, <font color="Maroon" title="c5">t2</font> be    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>







<b>set </b><font color="Maroon" title="c6">PTA</font> =  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c7">SPTA</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c4">t1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c4">t1</font></span>)</span> <b>by </b><i><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></i>;<br/>
<i><font color="Green" title="E51">A1</font></i>:  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c4">t1</font> = 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c8">x1</font>
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">R</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <span class="p3">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c4">t1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c8">x1</font>

;<br/>
<b>set </b><font color="Maroon" title="c9">CC1</font> =  <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c4">t1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">CR1</font> =  <a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <span class="p2">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c4">t1</font></span>)</span></span>)</span>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:75_2"/>
<font color="Maroon" title="c5">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c4">t1</font>
 ;<br/><a NAME="E2:75_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">t2</font>,<font color="Maroon" title="c8">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <span class="p2">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c4">t1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E2:75"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">s2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E4:75_2"/><i><font color="Green" title="E52">A2</font></i>: 
( <font color="Maroon" title="c11">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c4">t1</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">t2</font>,<font color="Maroon" title="c8">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s2</font> )
 <b>by </b><i><a class="ref" href="osalg_4.html#D11" title="OSALG_4:def.11">OSALG_4:def 11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">x11</font> = <font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c13">x22</font> = <font color="Maroon" title="c5">t2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s2</font> <b>by </b><i><a class="txt" href="osafree.html#E4:75_2"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>thus </b> <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c4">t1</font> = 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c12">x11</font>
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></i>
<br/>.= 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c13">x22</font>
<b>by </b><i><a class="txt" href="osafree.html#E4:75_2"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="osalg_4.html#T13" title="OSALG_4:th.13">OSALG_4:13</a></i>
<br/>.= 
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c5">t2</font>
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></i>
;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:75"/>
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c4">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c5">t2</font>
 ;<br/>

<b>hence </b><a NAME="E5:75"/>
<font color="Maroon" title="c5">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c4">t1</font>
 <b>by </b><i><a class="ref" href="osafree.html#T33" target="_self" title="OSAFREE:th.33">Th33</a></i>;<br/>


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