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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">O</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="lattice5.html#M1" target="_self">QuadrSeq</a> of <font color="Maroon">d</font>;<br/>









<a NAME="E1:53"/>
 <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#E2:53"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E2:53"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="lattice5.html#K8" target="_self">DistEsti</a> <font color="Maroon">d</font>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:53"/>
<font color="Maroon">q</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:53"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E5:53"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="lattice5.html#K8" target="_self">DistEsti</a> <font color="Maroon">d</font>, <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:53"><i><font color="Green">E39</font></i></a>, <a class="ref" href="wellord2.html#T22">WELLORD2:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">M</font> =  <a href="lattice5.html#K8" target="_self">DistEsti</a> <font color="Maroon">d</font> as   <a href="card_1.html#NM1">Cardinal</a> ;<br/>
<b>reconsider </b><font color="Maroon">N</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> as   <a href="card_1.html#NM1">Cardinal</a> <b>by </b><i/>;<br/>
<a NAME="E8:53"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T8">CARD_1:8</a></i>;<br/>
<b>hence </b><a NAME="E9:53"/>
( <font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K8" target="_self">DistEsti</a> <font color="Maroon">d</font> implies <font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> )
 ;<br/>

<b>assume </b><a NAME="E10:53"/>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font>
 ;<br/>

<b>hence </b><a NAME="E11:53"/>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K8" target="_self">DistEsti</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/>


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