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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">pai1</font>, <font color="Olive" title="b3">pai2</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Olive" title="b1">R</font><br/> for <font color="Olive" title="b4">k</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>   st <font color="Olive" title="b2">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">pai2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Olive" title="b2">pai1</font>,<font color="Olive" title="b3">pai2</font>,<font color="Olive" title="b4">k</font> is    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Olive" title="b1">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">pai1</font>, <font color="Olive" title="b2">pai2</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font><br/> for <font color="Olive" title="b3">k</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>   st <font color="Olive" title="b1">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">pai2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Olive" title="b1">pai1</font>,<font color="Olive" title="b2">pai2</font>,<font color="Olive" title="b3">k</font> is    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">pai1</font>, <font color="Maroon" title="c4">pai2</font> be    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</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>   st <font color="Maroon" title="c3">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pai2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Olive" title="b1">k</font> is    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pai2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font> is    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c6">pai</font> =  <a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>;<br/>
<span class="kw">assume </span><a NAME="E1:164"/><span class="lab"><font color="Green" title="E124">A1</font></span>: 
<font color="Maroon" title="c3">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pai2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font> is    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font></span><br/>

<a NAME="E2:164"/>
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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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#K2" title="NAT_1:func.2">+</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="164_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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#K2" title="NAT_1:func.2">+</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></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">n1</font> = <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:164_1_1"/>
( <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> or <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> or <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:164_1_1_1"/><span class="lab"><font color="Green" title="E125">A2</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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#K2" title="NAT_1:func.2">+</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></span><br/><div class="add"><a NAME="E2:164_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A3</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="lab"><font color="Green" title="E127">A4</font></span>: <span class="p1">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font> = 
 <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c7">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D69" target="_self" title="MODELC_1:def.69">Def69</a></span>
<br/>.= 
<font color="Maroon" title="c3">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:164_1_1_1"><span class="lab"><font color="Green" title="E126">A3</font></span></a>, <a class="ref" href="modelc_1.html#D68" target="_self" title="MODELC_1:def.68">Def68</a></span>
;<br/><span class="p1">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D69" target="_self" title="MODELC_1:def.69">Def69</a></span>
<br/>.= 
<font color="Maroon" title="c3">pai1</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164_1_1_1"><span class="lab"><font color="Green" title="E125">A2</font></span></a>, <a class="ref" href="modelc_1.html#D68" target="_self" title="MODELC_1:def.68">Def68</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:164_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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:164_1_1_1"><span class="lab"><font color="Green" title="E127">A4</font></span></a>, <a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:164_1_1_2"/><span class="lab"><font color="Green" title="E125">A5</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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#K2" title="NAT_1:func.2">+</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></span><br/><div class="add"><a NAME="E2:164_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A6</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="lab"><font color="Green" title="E127">A7</font></span>: <span class="p1">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font> = 
 <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c7">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D69" target="_self" title="MODELC_1:def.69">Def69</a></span>
<br/>.= 
<font color="Maroon" title="c3">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:164_1_1_2"><span class="lab"><font color="Green" title="E126">A6</font></span></a>, <a class="ref" href="modelc_1.html#D68" target="_self" title="MODELC_1:def.68">Def68</a></span>
;<br/><span class="p1">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D69" target="_self" title="MODELC_1:def.69">Def69</a></span>
<br/>.= 
<font color="Maroon" title="c4">pai2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164_1_1_2"><span class="lab"><font color="Green" title="E125">A5</font></span></a>, <a class="ref" href="modelc_1.html#D68" target="_self" title="MODELC_1:def.68">Def68</a></span>
<br/>.= 
<font color="Maroon" title="c3">pai1</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164"><span class="lab"><font color="Green" title="E124">A1</font></span></a>, <a class="txt" href="#E1:164_1_1_2"><span class="lab"><font color="Green" title="E125">A5</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:164_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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:164_1_1_2"><span class="lab"><font color="Green" title="E127">A7</font></span></a>, <a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:164_1_1_3"/><span class="lab"><font color="Green" title="E125">A8</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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#K2" title="NAT_1:func.2">+</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></span><br/><div class="add"><a NAME="E2:164_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A9</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">k</font> 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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/><span class="lab"><font color="Green" title="E127">A10</font></span>: <span class="p1">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D69" target="_self" title="MODELC_1:def.69">Def69</a></span>
<br/>.= 
<font color="Maroon" title="c4">pai2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164_1_1_3"><span class="lab"><font color="Green" title="E125">A8</font></span></a>, <a class="ref" href="modelc_1.html#D68" target="_self" title="MODELC_1:def.68">Def68</a></span>
<br/>.= 
<font color="Maroon" title="c4">pai2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/><span class="p1">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font> = 
 <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c7">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D69" target="_self" title="MODELC_1:def.69">Def69</a></span>
<br/>.= 
<font color="Maroon" title="c4">pai2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:164_1_1_3"><span class="lab"><font color="Green" title="E126">A9</font></span></a>, <a class="ref" href="modelc_1.html#D68" target="_self" title="MODELC_1:def.68">Def68</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:164_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font></span>)</span> <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#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:164_1_1_3"><span class="lab"><font color="Green" title="E127">A10</font></span></a>, <a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:164"/>
 <a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font> is    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
