<?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">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> 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="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>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Olive" title="b3">f</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#K55" title="MODELC_1:func.55">Fax</a> <font color="Olive" title="b3">f</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Olive" title="b3">f</font></span>)</span> )</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">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> 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="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>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Olive" title="b2">f</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#K55" title="MODELC_1:func.55">Fax</a> <font color="Olive" title="b2">f</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Olive" title="b2">f</font></span>)</span> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</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="b2">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="b2">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Olive" title="b1">f</font> iff <font color="Olive" title="b2">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="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Olive" title="b1">f</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</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#K55" title="MODELC_1:func.55">Fax</a> <font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font> iff <font color="Maroon" title="c5">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"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">g</font> =  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">thus </span><a NAME="E1:165"/>
( <font color="Maroon" title="c5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font> implies <font color="Maroon" title="c5">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"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span> )
  <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="c5">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"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <font color="Maroon" title="c5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="165_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c7">g</font> =  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">assume </span><a NAME="E1:165_1"/>
<font color="Maroon" title="c5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font>
 ; <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="c5">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"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:165_1"/><span class="lab"><font color="Green" title="E125">A1</font></span>: 
<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="c5">s</font>
 <span class="kw">and </span><br/><a NAME="E4:165_1"/><span class="lab"><font color="Green" title="E126">A2</font></span>: 
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  <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T15" target="_self" title="MODELC_1:th.15">Th15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">pai1</font> =  <a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c8">pai</font>,1;<br/>
<a NAME="E5:165_1"/><span class="lab"><font color="Green" title="E127">A3</font></span>: 
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">(<span class="default"><a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c8">pai</font>,1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</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="165_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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">(<span class="default"><a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c8">pai</font>,1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">n1</font> = <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<a NAME="E1:165_1_1"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c8">pai</font>,1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">n</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">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#D67" target="_self" title="MODELC_1:def.67">Def67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:165_1_1"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c8">pai</font>,1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:165_1"><span class="lab"><font color="Green" title="E126">A2</font></span></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>
<a NAME="E6:165_1"/>
<font color="Maroon" title="c8">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K57" title="MODELC_1:func.57">PathShift</a> <font color="Maroon" title="c8">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D67" target="_self" title="MODELC_1:def.67">Def67</a></span>;<br/>
<a NAME="E7:165_1"/><span class="kw">then </span>
<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>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:165_1"><span class="lab"><font color="Green" title="E127">A3</font></span></a>, <a class="ref" href="modelc_1.html#T15" target="_self" title="MODELC_1:th.15">Th15</a></span>;<br/>
<a NAME="E8:165_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A4</font></span>: 
<font color="Maroon" title="c5">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"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:165_1"><span class="lab"><font color="Green" title="E125">A1</font></span></a>, <a class="ref" href="modelc_1.html#T14" target="_self" title="MODELC_1:th.14">Th14</a></span>;<br/>
<a NAME="E9:165_1"/>
<font color="Maroon" title="c5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:165_1"><span class="lab"><font color="Green" title="E125">A1</font></span></a>, <a class="txt" href="#E4:165_1"><span class="lab"><font color="Green" title="E126">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:165_1"/>
<font color="Maroon" title="c5">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"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:165_1"><span class="lab"><font color="Green" title="E128">A4</font></span></a>, <a class="ref" href="modelc_1.html#T13" target="_self" title="MODELC_1:th.13">Th13</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>

<span class="kw">assume </span><a NAME="E2:165"/><span class="lab"><font color="Green" title="E125">A5</font></span>: 
<font color="Maroon" title="c5">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"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <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="c5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span><br/>

<a NAME="E3:165"/><span class="kw">then </span>
<font color="Maroon" title="c5">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"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T13" target="_self" title="MODELC_1:th.13">Th13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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><span class="kw"> such that </span><br/><a NAME="E5:165"/><span class="lab"><font color="Green" title="E126">A6</font></span>: 
<font color="Maroon" title="c7">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="c5">s</font>
 <span class="kw">and </span><br/><a NAME="E6:165"/><span class="lab"><font color="Green" title="E127">A7</font></span>: 
<font color="Maroon" title="c7">pai1</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>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T14" target="_self" title="MODELC_1:th.14">Th14</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">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><span class="kw"> such that </span><br/><a NAME="E8:165"/><span class="lab"><font color="Green" title="E128">A8</font></span>: 
<font color="Maroon" title="c8">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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1
 <span class="kw">and </span><br/><a NAME="E9:165"/><span class="lab"><font color="Green" title="E129">A9</font></span>: 
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  <font color="Maroon" title="c8">pai2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:165"><span class="lab"><font color="Green" title="E127">A7</font></span></a>, <a class="ref" href="modelc_1.html#T15" target="_self" title="MODELC_1:th.15">Th15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">pai</font> =  <a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c7">pai1</font>,<font color="Maroon" title="c8">pai2</font>,1;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">pai</font> =  <a href="modelc_1.html#K59" title="MODELC_1:func.59">PathConc</a> <font color="Maroon" title="c7">pai1</font>,<font color="Maroon" title="c8">pai2</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:165"><span class="lab"><font color="Green" title="E128">A8</font></span></a>, <a class="ref" href="modelc_1.html#T38" target="_self" title="MODELC_1:th.38">Th38</a></span>;<br/>
<span class="lab"><font color="Green" title="E130">A10</font></span>: <font color="Maroon" title="c10">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="c7">pai1</font>,<font color="Maroon" title="c8">pai2</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<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="c5">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:165"><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/>
<a NAME="E12:165"/>
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  <font color="Maroon" title="c10">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</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="165_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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"> <font color="Maroon" title="c10">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</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="165_3_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="E1:165_3_1"/>
( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or  not <font color="Maroon" title="c11">n</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="165_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:165_3_1_1"/><span class="lab"><font color="Green" title="E131">A11</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <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="c10">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</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"><a NAME="E2:165_3_1_1"/>
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="165_3_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:165_3_1_1_1"/><span class="lab"><font color="Green" title="E132">A12</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"> contradiction</span><br/>

<a NAME="E2:165_3_1_1_1"/>
<font color="Maroon" title="c11">n</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>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:165_3_1_1"><span class="lab"><font color="Green" title="E131">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:165_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:165_3_1_1_1"><span class="lab"><font color="Green" title="E132">A12</font></span></a>, <a class="ref" href="nat_1.html#T22" title="NAT_1:th.22">NAT_1:22</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><span class="kw">hence </span><a NAME="E3:165_3_1_1"/>
<font color="Maroon" title="c10">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:165"><span class="lab"><font color="Green" title="E125">A5</font></span></a>, <a class="txt" href="#E11:165"><span class="lab"><font color="Green" title="E130">A10</font></span></a>, <a class="ref" href="modelc_1.html#T13" target="_self" title="MODELC_1:th.13">Th13</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="165_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:165_3_1_2"/><span class="lab"><font color="Green" title="E131">A13</font></span>: 
not <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <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="c10">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</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"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">m</font> = <font color="Maroon" title="c11">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>  <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/><font color="Maroon" title="c10">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> = 
 <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c7">pai1</font>,<font color="Maroon" title="c8">pai2</font>,1,<font color="Maroon" title="c11">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="c8">pai2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:165_3_1_2"><span class="lab"><font color="Green" title="E131">A13</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="E4:165_3_1_2"/>
<font color="Maroon" title="c10">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:165"><span class="lab"><font color="Green" title="E129">A9</font></span></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="E13:165"/>
<font color="Maroon" title="c5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:165"><span class="lab"><font color="Green" title="E130">A10</font></span></a>, <a class="ref" href="modelc_1.html#T15" target="_self" title="MODELC_1:th.15">Th15</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>
