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<div class="add">

<b>let </b><font color="Maroon" title="c1">m</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> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E31">A1</font></i>: 
 <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:50"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">X'</font> = <font color="Maroon" title="c3">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <b>by </b><i><a class="ref" href="heyting3.html#T32" target="_self" title="HEYTING3:th.32">Th32</a></i>;<br/>
<b>assume </b><a NAME="E4:50"/><i><font color="Green" title="E33">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>   st <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/> not <font color="Olive" title="b1">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/>

<div><i><font color="Green" title="E34">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">k</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> ;<br/><b>reconsider </b><font color="Maroon" title="c6">Pk</font> =  <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c1">m</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="heyting3.html#T28" target="_self" title="HEYTING3:th.28">Th28</a></i>;<br/><a NAME="E2:50_1"/>
<font color="Maroon" title="c6">Pk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#D5" target="_self" title="HEYTING3:def.5">Def5</a></i>;<br/><a NAME="E3:50_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">Pk</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:50"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:50_1"/><i><font color="Green" title="E35">A5</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Pk</font> &amp; <font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="txt" href="heyting3.html#E2:50"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="heyting3.html#T15" target="_self" title="HEYTING3:th.15">Th15</a></i>;<br/><a NAME="E6:50_1"/><i><font color="Green" title="E36">A6</font></i>: 
(  ex <font color="Olive" title="b1">m'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>  st <br/>( <font color="Olive" title="b1">m'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Olive" title="b1">m'</font>,<font color="Maroon" title="c1">m</font> ) or <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c1">m</font> )
 <b>by </b><i><a class="txt" href="heyting3.html#E5:50_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="heyting3.html#D4" target="_self" title="HEYTING3:def.4">Def4</a></i>;<br/><a NAME="E7:50_1"/><i><font color="Green" title="E37">A7</font></i>: 
for <font color="Olive" title="b1">m'</font> being   <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>   holds <br/>(  not <font color="Olive" title="b1">m'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> or  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Olive" title="b1">m'</font>,<font color="Maroon" title="c1">m</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c8">m'</font> being    <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> <b> such that </b><a NAME="E2:50_1_1"/><i><font color="Green" title="E38">A8</font></i>: 
( <font color="Maroon" title="c8">m'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c8">m'</font>,<font color="Maroon" title="c1">m</font> )
 ;<br/>

<a NAME="E3:50_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">m'</font></span>)</span>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c8">m'</font>,<font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#D2" target="_self" title="HEYTING3:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:50_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E4:50"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="heyting3.html#E5:50_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="heyting3.html#E2:50_1_1"><i><font color="Green" title="E38">A8</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:50_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#D3" target="_self" title="HEYTING3:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E9:50_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E5:50_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="heyting3.html#E6:50_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="heyting3.html#E7:50_1"><i><font color="Green" title="E37">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">l</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> <b> such that </b><br/><a NAME="E7:50"/><i><font color="Green" title="E35">A9</font></i>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X'</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T6" target="_self" title="HEYTING3:th.6">Th6</a></i>;<br/>
<b>thus </b><a NAME="E8:50"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E5:50"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="heyting3.html#E7:50"><i><font color="Green" title="E35">A9</font></i></a></i>;<br/>


</div>
