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<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> &amp; ( for <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> ) &amp;  ex <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> st <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> &amp; ( for <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> ) implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:5_1_2"/><i><font color="Green" title="E3">A6</font></i>: 
 ex <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font>
 <b>and </b><br/><a NAME="E2:5_1_2"/><i><font color="Green" title="E4">A7</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font>
 <b>and </b><br/><a NAME="E3:5_1_2"/><i><font color="Green" title="E5">A8</font></i>: 
 ex <font color="Olive" title="b1">B</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> st <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">B</font>
 <b>and </b><br/><a NAME="E4:5_1_2"/><i><font color="Green" title="E6">A9</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">A</font> being    <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font><b> such that </b><br/><a NAME="E6:5_1_2"/><i><font color="Green" title="E7">A10</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">A</font> &amp; ( for <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> ) )
 <b>by </b><i><a class="txt" href="#E1:5_1_2"><i><font color="Green" title="E3">A6</font></i></a>, <a class="txt" href="#E2:5_1_2"><i><font color="Green" title="E4">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">B</font> being    <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font><b> such that </b><br/><a NAME="E8:5_1_2"/><i><font color="Green" title="E8">A11</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">B</font> &amp; ( for <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> ) )
 <b>by </b><i><a class="txt" href="#E3:5_1_2"><i><font color="Green" title="E5">A8</font></i></a>, <a class="txt" href="#E4:5_1_2"><i><font color="Green" title="E6">A9</font></i></a></i>;<br/>
<a NAME="E9:5_1_2"/><i><font color="Green" title="E9">A12</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="txt" href="#E6:5_1_2"><i><font color="Green" title="E7">A10</font></i></a>, <a class="txt" href="#E8:5_1_2"><i><font color="Green" title="E8">A11</font></i></a></i>;<br/>
<a NAME="E10:5_1_2"/>
<font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="#E6:5_1_2"><i><font color="Green" title="E7">A10</font></i></a>, <a class="txt" href="#E8:5_1_2"><i><font color="Green" title="E8">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:5_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="#E9:5_1_2"><i><font color="Green" title="E9">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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