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

<b>let </b><font color="Maroon" title="c1">S</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>let </b><font color="Maroon" title="c2">R</font> be  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">s0</font>, <font color="Maroon" title="c4">s1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>







<b>assume </b><a NAME="E1:142"/><i><font color="Green" title="E108">A1</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">s0</font>,<font color="Maroon" title="c4">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">pai1</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font><b> such that </b><br/><a NAME="E3:142"/><i><font color="Green" title="E109">A2</font></i>: 
<font color="Maroon" title="c5">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#T25" target="_self" title="MODELC_1:th.25">Th25</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> =  <a href="modelc_1.html#K51" title="MODELC_1:func.51">PrePath</a> $1,<font color="Maroon" title="c3">s0</font>,<font color="Maroon" title="c5">pai1</font>;<br/>
<a NAME="E4:142"/><i><font color="Green" title="E110">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c6">pai</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E6:142"/><i><font color="Green" title="E111">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c6">pai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="modelc_1.html#E4:142"><i><font color="Green" title="E110">A3</font></i></a>);<br/></i>
<a NAME="E7:142"/>
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>  holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n</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>set </b><font color="Maroon" title="c8">n1</font> = <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>set </b><font color="Maroon" title="c9">n0</font> = <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:142_1_1"/>
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_1"><b>suppose </b></a><a NAME="E1:142_1_1_1"/><i><font color="Green" title="E112">A5</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><i><font color="Green" title="E113">A6</font></i>: <font color="Maroon" title="c6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">n</font>)
<b>by </b><i><a class="txt" href="modelc_1.html#E6:142"><i><font color="Green" title="E111">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">s0</font>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:142_1_1_1"><i><font color="Green" title="E112">A5</font></i></a>, <a class="ref" href="modelc_1.html#D61" target="_self" title="MODELC_1:def.61">Def61</a></i>
;<br/><i><font color="Green" title="E114">A7</font></i>:  <a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> = 
 <a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a>, <a class="txt" href="modelc_1.html#E1:142_1_1_1"><i><font color="Green" title="E112">A5</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
;<br/><font color="Maroon" title="c6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
<b>by </b><i><a class="txt" href="modelc_1.html#E6:142"><i><font color="Green" title="E111">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">s1</font>
<b>by </b><i><a class="txt" href="modelc_1.html#E3:142_1_1_1"><i><font color="Green" title="E114">A7</font></i></a>, <a class="ref" href="modelc_1.html#D61" target="_self" title="MODELC_1:def.61">Def61</a>, <a class="txt" href="modelc_1.html#E3:142"><i><font color="Green" title="E109">A2</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:142_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E2:142_1_1_1"><i><font color="Green" title="E113">A6</font></i></a>, <a class="txt" href="modelc_1.html#E1:142"><i><font color="Green" title="E108">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_2"><b>suppose </b></a><a NAME="E1:142_1_1_2"/><i><font color="Green" title="E112">A8</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:142_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">n0</font> = <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>by </b><i><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/><i><font color="Green" title="E113">A9</font></i>: <font color="Maroon" title="c6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">n</font>)
<b>by </b><i><a class="txt" href="modelc_1.html#E6:142"><i><font color="Green" title="E111">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:142_1_1_2"><i><font color="Green" title="E112">A8</font></i></a>, <a class="ref" href="modelc_1.html#D61" target="_self" title="MODELC_1:def.61">Def61</a></i>
<br/>.= 
<font color="Maroon" title="c5">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c5">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">n0</font>
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
;<br/><font color="Maroon" title="c6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
<b>by </b><i><a class="txt" href="modelc_1.html#E6:142"><i><font color="Green" title="E111">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D61" target="_self" title="MODELC_1:def.61">Def61</a></i>
<br/>.= 
<font color="Maroon" title="c5">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c5">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
;<br/><b>hence </b><a NAME="E6:142_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E4:142_1_1_2"><i><font color="Green" title="E113">A9</font></i></a>, <a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">pai</font> = <font color="Maroon" title="c6">pai</font> as    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> <b>by </b><i><a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def39</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">pai</font>
;<br/>

<i><font color="Green" title="E112">A10</font></i>: <font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 = 
<font color="Maroon">H<sub>1</sub></font>(0)
<b>by </b><i><a class="txt" href="modelc_1.html#E6:142"><i><font color="Green" title="E111">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">s0</font>
<b>by </b><i><a class="ref" href="modelc_1.html#D61" target="_self" title="MODELC_1:def.61">Def61</a></i>
;<br/>
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 = 
<font color="Maroon">H<sub>1</sub></font>(1)
<b>by </b><i><a class="txt" href="modelc_1.html#E6:142"><i><font color="Green" title="E111">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D61" target="_self" title="MODELC_1:def.61">Def61</a></i>
<br/>.= 
<font color="Maroon" title="c5">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c4">s1</font>
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a>, <a class="txt" href="modelc_1.html#E3:142"><i><font color="Green" title="E109">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E11:142"/>
( <font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s0</font> &amp; <font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> )
 <b>by </b><i><a class="txt" href="modelc_1.html#E9:142"><i><font color="Green" title="E112">A10</font></i></a></i>;<br/>


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