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<b>let </b><font color="Maroon">S</font> be  <a href="osalg_4.html#V2" target="_self">locally_directed</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">U1</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">E</font> be  <a href="msualg_4.html#V3">MSEquivalence-like</a>  <a href="osalg_4.html#M1" target="_self">OrderSortedRelation</a> of <font color="Maroon">U1</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/>









<b>reconsider </b><font color="Maroon">SU</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> as   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>
<a NAME="E2:40"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="osalg_4.html#K4" target="_self">-carrier_of</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:40"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:40"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">E</font> is <a href="osalg_4.html#V1" target="_self">os-compatible</a>
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_1"/>
 <a href="osalg_4.html#K8" target="_self">OSClass</a> <font color="Maroon">E</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K8" target="_self">OSClass</a> <font color="Maroon">E</font>,<font color="Maroon">y</font>
 ;<br/><a NAME="E2:40_1"/><b>then </b>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">E</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">E</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Maroon">y</font>
 ;<br/><a NAME="E3:40_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">E</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/><b>then consider </b><font color="Maroon">s1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:40_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s1</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">sn1</font> = <font color="Maroon">s</font>, <font color="Maroon">s11</font> = <font color="Maroon">s1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/><a NAME="E7:40_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SU</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s11</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SU</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s11</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>consider </b><font color="Maroon">s2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E9:40_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s11</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">sn1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/><a NAME="E10:40_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font>
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E11:40_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E6:40"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E7:40"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="osalg_4.html#K4" target="_self">-carrier_of</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:40"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:40"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">E</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:40"/><b>then </b>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">E</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">E</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>
<b>hence </b><a NAME="E11:40"/>
 <a href="osalg_4.html#K8" target="_self">OSClass</a> <font color="Maroon">E</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K8" target="_self">OSClass</a> <font color="Maroon">E</font>,<font color="Maroon">y</font>
 ;<br/>


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