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<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">k</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
(  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">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> ) or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font> )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/>
 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">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">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> <b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">m</font>,<font color="Maroon">k</font> )
 ;<br/><b>thus </b><a NAME="E4:39_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> = <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E2:39_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font>
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E3:39_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> =  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font>;<br/><a NAME="E2:39_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:39_1_1_1_1_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</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/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> =  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E2:39_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:39_1_2"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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