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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">A</font> be   <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">A</font>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">s2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>







<b>reconsider </b><font color="Maroon">s3</font> = <font color="Maroon">s1</font>, <font color="Maroon">s4</font> = <font color="Maroon">s2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>assume </b><a NAME="E2:35"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font>
 ;<br/>

<a NAME="E3:35"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s2</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:35"/>
 <a href="osalg_4.html#K6" target="_self">OSClass</a> <font color="Maroon">E</font>,<font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="osalg_4.html#K6" target="_self">OSClass</a> <font color="Maroon">E</font>,<font color="Maroon">s2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K6" target="_self">OSClass</a> <font color="Maroon">E</font>,<font color="Maroon">s1</font>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green">E45</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">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">z</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">s1</font></span>)</span></span>)</span>,<font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">SO</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</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="E5:35_1"/>
<font color="Maroon">SO</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">SO</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s4</font>
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#D18">OSALG_1:def 18</a></i>;<br/>
<b>hence </b><a NAME="E6:35_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K6" target="_self">OSClass</a> <font color="Maroon">E</font>,<font color="Maroon">s2</font>
 <b>by </b><i>, , </i>;<br/>


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


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