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<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><a NAME="E1:43"/>
(  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="tarski.html#R1">c=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">k</font>,<font color="Maroon">m</font> implies <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E30</font></i>: 
 <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="tarski.html#R1">c=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">k</font>,<font color="Maroon">m</font>
 ;<br/>

<b>assume </b><a NAME="E2:43_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:43_1"/><i><font color="Green">E46</font></i>: 
<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">n</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">n</font>,<font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:43_1"/>
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">n</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#K2" target="_self">PFArt</a> <font color="Maroon">k</font>,<font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1"/><i><font color="Green">E48</font></i>: 
<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">n</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#K2" target="_self">PFArt</a> <font color="Maroon">k</font>,<font color="Maroon">m</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_1_1"/>
(  ex <font color="Olive">m'</font> being <a href="abian.html#V1">odd</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> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">m'</font>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <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">n</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> ) or <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">k</font></span>)</span>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <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">n</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> )
 </span><b>by </b><i><a class="txt" href="heyting3.html#E2:43_1"><i><font color="Green">E31</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1_1"/>
 ex <font color="Olive">m'</font> being <a href="abian.html#V1">odd</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> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">m'</font>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <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">n</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> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m'</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:43_1_1_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">m'</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m'</font>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <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">n</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> )
 ;<br/><a NAME="E4:43_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E5:43_1_1_1_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T20">NAT_1:20</a></i>;<br/><b>hence </b><a NAME="E6:43_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_1_2"/>
<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">k</font></span>)</span>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <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">n</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>
 ;<br/><div class="add"><a NAME="E2:43_1_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>thus </b><a NAME="E3:43_1_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:43_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="heyting3.html#E1:43_1"><i><font color="Green">E30</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:43"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> implies  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="tarski.html#R1">c=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">k</font>,<font color="Maroon">m</font> )
 <b>by </b><i/>;<br/>


</div>
