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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">z</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">x</font> =  <a href="osalg_1.html#K4" title="OSALG_1:func.4">ConstOSSet</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">z</font>;<br/>
<b>set </b><font color="Maroon" title="c4">D</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <font color="Maroon" title="c2">z</font>;<br/>
<a NAME="E1:50"/>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> holds <br/> not <span class="p1">(<span class="default"><a href="osalg_1.html#K4" title="OSALG_1:func.4">ConstOSSet</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E2:50"/>
 <a href="osalg_1.html#K4" title="OSALG_1:func.4">ConstOSSet</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">z</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 <b>by </b><i><a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>

<b>assume </b><a NAME="E3:50"/>
<font color="Maroon" title="c5">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">s2</font>
 ;<br/>

<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s1</font> = 
<font color="Maroon" title="c2">z</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">s2</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E5:50"/>
<span class="p1">(<span class="default"><a href="osalg_1.html#K4" title="OSALG_1:func.4">ConstOSSet</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K4" title="OSALG_1:func.4">ConstOSSet</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font>
 ;<br/>


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