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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</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>thus </b><a NAME="E1:42"/>
(  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">m</font> implies <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E25">A1</font></i>: 
 <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">m</font>
 ;<br/>

<b>assume </b><a NAME="E2:42_1"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E3:42_1"/><i><font color="Green" title="E27">A3</font></i>: 
<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="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">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="c1">n</font>,<font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#D3" target="_self" title="HEYTING3:def.3">Def3</a></i>;<br/>
<a NAME="E4:42_1"/>
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="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">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="c3">k</font>,<font color="Maroon" title="c2">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1_1"/><i><font color="Green" title="E28">A4</font></i>: 
<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="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">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="c3">k</font>,<font color="Maroon" title="c2">m</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_1"/>
(  ex <font color="Olive" title="b1">m'</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">m'</font>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) or <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="c3">k</font></span>)</span>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 </span><b>by </b><i><a class="txt" href="heyting3.html#E1:42_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="heyting3.html#D2" target="_self" title="HEYTING3:def.2">Def2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1_1"/>
 ex <font color="Olive" title="b1">m'</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">m'</font>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">m'</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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="E3:42_1_1_1_1"/><i><font color="Green" title="E29">A5</font></i>: 
( <font color="Maroon" title="c4">m'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">m'</font>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 ;<br/><a NAME="E4:42_1_1_1_1"/><i><font color="Green" title="E30">A6</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:42_1_1_1_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E5:42_1_1_1_1"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:42_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></i>;<br/><b>hence </b><a NAME="E6:42_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E4:42_1_1_1_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_1_2"/>
<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="c3">k</font></span>)</span>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:42_1"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E1:42_1"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="heyting3.html#E3:42_1"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:42"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> implies  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="txt" href="heyting3.html#E33"><i><font color="Green" title="E20">Lm7</font></i></a></i>;<br/>


</div>
