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<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:5_1_2"/><i><font color="Green">E41</font></i>: 
 ex <font color="Olive">A</font> being   <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">A</font>
 <b>and </b><br/><a NAME="E2:5_1_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">A</font> being   <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font> holds   <a href="card_1.html#K4">card</a> <font color="Olive">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E3:5_1_2"/><i><font color="Green">E49</font></i>: 
 ex <font color="Olive">B</font> being   <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font> st <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">B</font>
 <b>and </b><br/><a NAME="E4:5_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">A</font> being   <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font> holds   <a href="card_1.html#K4">card</a> <font color="Olive">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/>

<b>consider </b><font color="Maroon">A</font> being    <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font><b> such that </b><br/><a NAME="E6:5_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">A</font> being   <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font> holds   <a href="card_1.html#K4">card</a> <font color="Olive">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">B</font> being    <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font><b> such that </b><br/><a NAME="E8:5_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">A</font> being   <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font> holds   <a href="card_1.html#K4">card</a> <font color="Olive">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:5_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:5_1_2"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:5_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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