<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">BASSIGN</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span><br/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being  <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Olive" title="b1">R</font>,<font color="Olive" title="b2">BASSIGN</font></span>)</span><br/> for <font color="Olive" title="b5">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b3">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Olive" title="b4">g</font> iff <font color="Olive" title="b5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b3">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Olive" title="b4">g</font></span>)</span> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">BASSIGN</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c2">R</font>,<font color="Olive" title="b1">BASSIGN</font></span>)</span><br/> for <font color="Olive" title="b4">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b4">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b2">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Olive" title="b3">g</font> iff <font color="Olive" title="b4">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b2">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Olive" title="b3">g</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">BASSIGN</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span><br/> for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b1">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Olive" title="b2">g</font> iff <font color="Olive" title="b3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Olive" title="b2">g</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font> iff <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c6">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font> iff <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> )</span><br/>









<a NAME="E1:175"/><i><font color="Green" title="E132">A1</font></i>: 
( <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font> implies <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:175_1"/>
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c7">pai</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="E3:175_1"/><i><font color="Green" title="E133">A2</font></i>: 
<font color="Maroon" title="c7">pai</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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 <b>and </b><br/><a NAME="E4:175_1"/><i><font color="Green" title="E134">A3</font></i>: 
 ex <font color="Olive" title="b1">m</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 <br/>( ( for <font color="Olive" title="b2">j</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">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> ) &amp; <font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font> )
 <b>by </b><i><a class="ref" href="modelc_1.html#T16" target="_self" title="MODELC_1:th.16">Th16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">m</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> <b> such that </b><br/><a NAME="E6:175_1"/><i><font color="Green" title="E135">A4</font></i>: 
for <font color="Olive" title="b1">j</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">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font> holds <br/><font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E7:175_1"/><i><font color="Green" title="E136">A5</font></i>: 
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E4:175_1"><i><font color="Green" title="E134">A3</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="175_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:175_1_1"/>
( <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="175_1_1_1"><b>suppose </b></a><a NAME="E1:175_1_1_1"/>
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:175_1_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:175_1"><i><font color="Green" title="E133">A2</font></i></a>, <a class="txt" href="modelc_1.html#E7:175_1"><i><font color="Green" title="E136">A5</font></i></a>, <a class="ref" href="modelc_1.html#T17" target="_self" title="MODELC_1:th.17">Th17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="175_1_1_2"><b>suppose </b></a><a NAME="E1:175_1_1_2"/><i><font color="Green" title="E137">A6</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span></span><br/><div class="add"><b>set </b><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font>;<br/><b>set </b><font color="Maroon" title="c10">k</font> = <font color="Maroon" title="c8">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/><b>reconsider </b><font color="Maroon" title="c11">k</font> = <font color="Maroon" title="c8">m</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="txt" href="modelc_1.html#E1:175_1_1_2"><i><font color="Green" title="E137">A6</font></i></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/><a NAME="E3:175_1_1_2"/><i><font color="Green" title="E138">A7</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:175_1"><i><font color="Green" title="E133">A2</font></i></a>, <a class="txt" href="modelc_1.html#E6:175_1"><i><font color="Green" title="E135">A4</font></i></a>, <a class="txt" href="modelc_1.html#E1:175_1_1_2"><i><font color="Green" title="E137">A6</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c12">pai1</font> =  <a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c7">pai</font>,1;<br/><i><font color="Green" title="E139">A8</font></i>: <span class="p1">(<span class="default"><a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c7">pai</font>,1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D67" target="_self" title="MODELC_1:def.67">Def67</a></i>
<br/>.= 
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1

;<br/><i><font color="Green" title="E140">A9</font></i>: <span class="p1">(<span class="default"><a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c7">pai</font>,1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> = 
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D67" target="_self" title="MODELC_1:def.67">Def67</a></i>
<br/>.= 
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font>

;<br/><a NAME="E6:175_1_1_2"/>
for <font color="Olive" title="b1">j</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">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c7">pai</font>,1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">j</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font> implies <span class="p1">(<span class="default"><a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c7">pai</font>,1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<b>assume </b><a NAME="E1:175_1_1_2_3"/><i><font color="Green" title="E141">A10</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c7">pai</font>,1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font></span><br/>

<b>set </b><font color="Maroon" title="c14">j1</font> = <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<a NAME="E2:175_1_1_2_3"/>
<font color="Maroon" title="c13">j</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="c8">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1_1_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:175_1_1_2_3_1"/>
<font color="Maroon" title="c13">j</font> <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> <font color="Maroon" title="c11">k</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:175_1_1_2_3"><i><font color="Green" title="E141">A10</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/>
<a NAME="E2:175_1_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c13">j</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="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:175_1_1_2_3_1"/>
<font color="Maroon" title="c13">j</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="c8">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:175_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E6:175_1"><i><font color="Green" title="E135">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:175_1_1_2_3"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c7">pai</font>,1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D67" target="_self" title="MODELC_1:def.67">Def67</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E7:175_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E7:175_1"><i><font color="Green" title="E136">A5</font></i></a>, <a class="txt" href="modelc_1.html#E4:175_1_1_2"><i><font color="Green" title="E139">A8</font></i></a>, <a class="txt" href="modelc_1.html#E5:175_1_1_2"><i><font color="Green" title="E140">A9</font></i></a>, <a class="ref" href="modelc_1.html#T16" target="_self" title="MODELC_1:th.16">Th16</a></i>;<br/><a NAME="E8:175_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:175_1"><i><font color="Green" title="E133">A2</font></i></a>, <a class="ref" href="modelc_1.html#T14" target="_self" title="MODELC_1:th.14">Th14</a></i>;<br/><a NAME="E9:175_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K55" title="MODELC_1:func.55">Fax</a> <font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:175_1_1_2"><i><font color="Green" title="E138">A7</font></i></a>, <a class="ref" href="modelc_1.html#T13" target="_self" title="MODELC_1:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E10:175_1_1_2"/>
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="modelc_1.html#T17" target="_self" title="MODELC_1:th.17">Th17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:175"/>
( <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> implies <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:175_2"/><i><font color="Green" title="E133">A11</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="175_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:175_2_1"/>
( <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font> or <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K55" title="MODELC_1:func.55">Fax</a> <font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="modelc_1.html#E1:175_2"><i><font color="Green" title="E133">A11</font></i></a>, <a class="ref" href="modelc_1.html#T17" target="_self" title="MODELC_1:th.17">Th17</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="175_2_1_1"><b>suppose </b></a><a NAME="E1:175_2_1_1"/><i><font color="Green" title="E134">A12</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span><br/><div class="add"><b>consider </b><font color="Maroon" title="c7">pai</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="E3:175_2_1_1"/><i><font color="Green" title="E135">A13</font></i>: 
<font color="Maroon" title="c7">pai</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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</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>set </b><font color="Maroon" title="c8">m</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><a NAME="E4:175_2_1_1"/>
for <font color="Olive" title="b1">j</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">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 ;<br/><b>hence </b><a NAME="E5:175_2_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:175_2_1_1"><i><font color="Green" title="E134">A12</font></i></a>, <a class="txt" href="modelc_1.html#E3:175_2_1_1"><i><font color="Green" title="E135">A13</font></i></a>, <a class="ref" href="modelc_1.html#T16" target="_self" title="MODELC_1:th.16">Th16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="175_2_1_2"><b>suppose </b></a><a NAME="E1:175_2_1_2"/><i><font color="Green" title="E134">A14</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K55" title="MODELC_1:func.55">Fax</a> <font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span><br/><div class="add"><b>set </b><font color="Maroon" title="c7">h</font> = <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font>;<br/><a NAME="E2:175_2_1_2"/><i><font color="Green" title="E135">A15</font></i>: 
( <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:175_2_1_2"><i><font color="Green" title="E134">A14</font></i></a>, <a class="ref" href="modelc_1.html#T13" target="_self" title="MODELC_1:th.13">Th13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">pai</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="E4:175_2_1_2"/><i><font color="Green" title="E136">A16</font></i>: 
<font color="Maroon" title="c8">pai</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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 <b>and </b><br/><a NAME="E5:175_2_1_2"/><i><font color="Green" title="E137">A17</font></i>: 
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#T14" target="_self" title="MODELC_1:th.14">Th14</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">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="E7:175_2_1_2"/><i><font color="Green" title="E138">A18</font></i>: 
<font color="Maroon" title="c9">pai1</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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1
 <b>and </b><br/><a NAME="E8:175_2_1_2"/><i><font color="Green" title="E139">A19</font></i>: 
 ex <font color="Olive" title="b1">m</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 <br/>( ( for <font color="Olive" title="b2">j</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">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c9">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> ) &amp; <font color="Maroon" title="c9">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font> )
 <b>by </b><i><a class="txt" href="modelc_1.html#E5:175_2_1_2"><i><font color="Green" title="E137">A17</font></i></a>, <a class="ref" href="modelc_1.html#T16" target="_self" title="MODELC_1:th.16">Th16</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">m</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> <b> such that </b><br/><a NAME="E10:175_2_1_2"/><i><font color="Green" title="E140">A20</font></i>: 
for <font color="Olive" title="b1">j</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">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> holds <br/><font color="Maroon" title="c9">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E11:175_2_1_2"/><i><font color="Green" title="E141">A21</font></i>: 
<font color="Maroon" title="c9">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E8:175_2_1_2"><i><font color="Green" title="E139">A19</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c11">PAI</font> =  <a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c8">pai</font>,<font color="Maroon" title="c9">pai1</font>,1;<br/><b>reconsider </b><font color="Maroon" title="c12">PAI</font> =  <a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c8">pai</font>,<font color="Maroon" title="c9">pai1</font>,1 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="txt" href="modelc_1.html#E7:175_2_1_2"><i><font color="Green" title="E138">A18</font></i></a>, <a class="ref" href="modelc_1.html#T38" target="_self" title="MODELC_1:th.38">Th38</a></i>;<br/><b>set </b><font color="Maroon" title="c13">m1</font> = <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><i><font color="Green" title="E142">A22</font></i>: <font color="Maroon" title="c12">PAI</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 href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c8">pai</font>,<font color="Maroon" title="c9">pai1</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="modelc_1.html#D69" target="_self" title="MODELC_1:def.69">Def69</a></i>
<br/>.= 
<font color="Maroon" title="c6">s</font>
<b>by </b><i><a class="txt" href="modelc_1.html#E4:175_2_1_2"><i><font color="Green" title="E136">A16</font></i></a>, <a class="ref" href="modelc_1.html#D68" target="_self" title="MODELC_1:def.68">Def68</a></i>
;<br/><a NAME="E14:175_2_1_2"/><i><font color="Green" title="E143">A23</font></i>: 
not <font color="Maroon" title="c10">m</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> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><i><font color="Green" title="E144">A24</font></i>: <font color="Maroon" title="c12">PAI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</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="c8">pai</font>,<font color="Maroon" title="c9">pai1</font>,1,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D69" target="_self" title="MODELC_1:def.69">Def69</a></i>
<br/>.= 
<font color="Maroon" title="c9">pai1</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="c10">m</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> 1</span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E14:175_2_1_2"><i><font color="Green" title="E143">A23</font></i></a>, <a class="ref" href="modelc_1.html#D68" target="_self" title="MODELC_1:def.68">Def68</a></i>
<br/>.= 
<font color="Maroon" title="c9">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font>

;<br/><a NAME="E16:175_2_1_2"/>
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="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><font color="Maroon" title="c12">PAI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_2_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c12">PAI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<b>assume </b><a NAME="E1:175_2_1_2_3"/><i><font color="Green" title="E145">A25</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">PAI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="175_2_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:175_2_1_2_3_1"/>
( <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or  not <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="175_2_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:175_2_1_2_3_1_1"/>
<font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">PAI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><b>hence </b><a NAME="E2:175_2_1_2_3_1_1"/>
<font color="Maroon" title="c12">PAI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E2:175_2_1_2"><i><font color="Green" title="E135">A15</font></i></a>, <a class="txt" href="modelc_1.html#E13:175_2_1_2"><i><font color="Green" title="E142">A22</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="175_2_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:175_2_1_2_3_1_2"/><i><font color="Green" title="E146">A26</font></i>: 
not <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">PAI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><b>set </b><font color="Maroon" title="c15">k0</font> = <font color="Maroon" title="c14">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/><b>reconsider </b><font color="Maroon" title="c16">k0</font> = <font color="Maroon" title="c14">k</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="txt" href="modelc_1.html#E1:175_2_1_2_3_1_2"><i><font color="Green" title="E146">A26</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/><i><font color="Green" title="E147">A27</font></i>: <font color="Maroon" title="c12">PAI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font> = 
 <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c8">pai</font>,<font color="Maroon" title="c9">pai1</font>,1,<font color="Maroon" title="c14">k</font>
<b>by </b><i><a class="ref" href="modelc_1.html#D69" target="_self" title="MODELC_1:def.69">Def69</a></i>
<br/>.= 
<font color="Maroon" title="c9">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">k0</font>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:175_2_1_2_3_1_2"><i><font color="Green" title="E146">A26</font></i></a>, <a class="ref" href="modelc_1.html#D68" target="_self" title="MODELC_1:def.68">Def68</a></i>
;<br/><a NAME="E4:175_2_1_2_3_1_2"/>
<font color="Maroon" title="c16">k0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_2_1_2_3_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:175_2_1_2_3_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:175_2_1_2_3"><i><font color="Green" title="E145">A25</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E2:175_2_1_2_3_1_2_2"/>
<font color="Maroon" title="c16">k0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:175_2_1_2_3_1_2"/>
<font color="Maroon" title="c12">PAI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E10:175_2_1_2"><i><font color="Green" title="E140">A20</font></i></a>, <a class="txt" href="modelc_1.html#E3:175_2_1_2_3_1_2"><i><font color="Green" title="E147">A27</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E17:175_2_1_2"/>
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E11:175_2_1_2"><i><font color="Green" title="E141">A21</font></i></a>, <a class="txt" href="modelc_1.html#E13:175_2_1_2"><i><font color="Green" title="E142">A22</font></i></a>, <a class="txt" href="modelc_1.html#E15:175_2_1_2"><i><font color="Green" title="E144">A24</font></i></a>, <a class="ref" href="modelc_1.html#T16" target="_self" title="MODELC_1:th.16">Th16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:175"/>
( <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font> iff <font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:175"><i><font color="Green" title="E132">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
