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<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A</font>,<font color="Maroon">L</font>;<br/>





<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K5">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K5">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">x'</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K5">]</a></span>
 ;<br/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">x'</font> = 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
;<br/>
<a NAME="E5:44"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K5">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:44"/>
 <a href="lattice5.html#K8" target="_self">DistEsti</a> <font color="Maroon">d</font>,<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K5">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:44"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:44"/>
 <a href="lattice5.html#K8" target="_self">DistEsti</a> <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="card_1.html#T46">CARD_1:46</a></i>;<br/>


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