<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">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/>

<a NAME="E1:29"/><i><font color="Green" title="E17">A1</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> 3
 
;<br/>
<b>thus </b><a NAME="E2:29"/>
 <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> 3</span>)</span>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font>
 ;<br/>

<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="E3:29_1"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &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">k</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> <font color="Maroon" title="c3">x</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#D3" target="_self" title="HEYTING3:def.3">Def3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1"/>
( <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="heyting3.html#E3:29_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1"/>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 or <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E3:29_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <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">k</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:29_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="heyting3.html#D3" target="_self" title="HEYTING3:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> 3</span>)</span>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2"/>
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:29_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> 3</span>)</span>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:29_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> 3</span>)</span>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:29"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> 3</span>)</span>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2"/>
( <font color="Maroon" title="c3">x</font> <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">k</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> 3</span>)</span>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="heyting3.html#E3:29"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1"><b>suppose </b></a><a NAME="E1:29_2_1"/>
<font color="Maroon" title="c3">x</font> <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">k</font>
 ;<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="E3:29_2_1"/><i><font color="Green" title="E19">A4</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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 &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">k</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> <font color="Maroon" title="c3">x</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#D3" target="_self" title="HEYTING3:def.3">Def3</a></i>;<br/><a NAME="E4:29_2_1"/>
<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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="heyting3.html#E26"><i><font color="Green" title="E15">Lm4</font></i></a></i>;<br/><a NAME="E5:29_2_1"/><b>then </b>
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="heyting3.html#E3:29_2_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:29_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:29_2_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="heyting3.html#D3" target="_self" title="HEYTING3:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2"><b>suppose </b></a><a NAME="E1:29_2_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> 3</span>)</span>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_2_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <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> 3</span>)</span>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:29_2_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="heyting3.html#T17" target="_self" title="HEYTING3:th.17">Th17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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