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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">m</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>



<b>assume </b><a NAME="E1:51"/><i><font color="Green">E30</font></i>: 
 <a href="heyting3.html#K5" target="_self">PFDrt</a> <font color="Maroon">m</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">a</font>
 ;<br/>

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:51"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">m</font></span><a href="domain_1.html#K6">}</a></span></span><a href="mcart_1.html#K12">:]</a></span> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:51"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">n</font>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> not <font color="Olive">n</font> is <a href="abian.html#V1">even</a>
 ;<br/>

<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>reconsider </b><font color="Maroon">Pk</font> =  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">k</font>,<font color="Maroon">m</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">m</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <b>by </b><i/>;<br/><a NAME="E2:51_1"/>
<font color="Maroon">Pk</font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K5" target="_self">PFDrt</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E3:51_1"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Pk</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:51_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pk</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:51_1"/><i><font color="Green">E50</font></i>: 
(  ex <font color="Olive">m'</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m'</font>,<font color="Maroon">m</font> ) or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">k</font>,<font color="Maroon">m</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:51_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">m'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not <font color="Olive">m'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or  not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m'</font>,<font color="Maroon">m</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">m'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:51_1_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">m'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">m'</font>,<font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:51_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m'</font></span>)</span>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">m'</font>,<font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:51_1_1"/><b>then </b>
not 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m'</font> is <a href="abian.html#V1">even</a>
 
<b>by </b><i>, , <a class="txt" href="heyting3.html#E5"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:51_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E8:51_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">k</font>,<font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:51_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="heyting3.html#E2"><i><font color="Green">Lemma39</font></i></a>, <a class="txt" href="heyting3.html#E3"><i><font color="Green">Lemma40</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">l</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:51"/><i><font color="Green">E95</font></i>: 
not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E8:51"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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