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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">R</font>, <font color="Maroon">Y</font> be   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">R</font>;<br/>



<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">X</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">Y</font> as   <a href="struct_0.html#NM8">ManySortedSet</a> of <font color="Maroon">R</font> ;<br/>
<a NAME="E2:1"/>
<font color="Maroon">M</font> is <a href="osalg_1.html#V11">order-sorted</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">s2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D18">OSALG_1:def 18</a><br/>

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font>
 ;<br/>

<a NAME="E2:1_1"/><i><font color="Green">E26</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>;<br/>
<a NAME="E3:1_1"/>
( <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</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="E4:1_1"/>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:1"/>
<font color="Maroon">X</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">Y</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">R</font>
 ;<br/>


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