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

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">N</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>, <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E18">A1</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b3">b</font> ) holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">b</font>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green" title="E19">A2</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p'</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( ( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> implies <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> ) &amp; ( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> implies <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> ) &amp; ( <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> implies <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> ) &amp; ( <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> implies <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> ) ) ) holds <br/>( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">b</font> iff <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p'</font>,<font color="Olive" title="b4">b</font> )
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green" title="E20">A3</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>] <b>means </b>( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $2 holds <br/>$2 <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $1,<font color="Olive" title="b1">a</font> or for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $2 holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $1,$2 );<br/>
<a NAME="E4:23"/><i><font color="Green" title="E21">A4</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">b</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span> ex <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">b</font>] ) &amp;  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">b</font>] )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">p</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>, <font color="Maroon" title="c5">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:23_1"/><i><font color="Green" title="E22">A5</font></i>: 
(  ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Olive" title="b1">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> ) &amp;  ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">c</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">b</font>] ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c6">a'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E5:23_1"/><i><font color="Green" title="E23">A6</font></i>: 
( <font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E3:23_1"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">c'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E7:23_1"/><i><font color="Green" title="E24">A7</font></i>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c'</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c7">c'</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E3:23_1"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E8:23_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b2">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_1"/>
( <font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c'</font> or <font color="Maroon" title="c6">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1_1"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c'</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c6">a'</font>
;<br/><b>take </b>
<font color="Maroon" title="c7">c'</font>
;<br/><b>thus </b><a NAME="E2:23_1_1_1_1"/>
( <font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c'</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c'</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c7">c'</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E5:23_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="arrow.html#E7:23_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="arrow.html#E1:23_1_1_1_1"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_1_2"/><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c6">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c'</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c8">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:23_1_1_1_2"/><i><font color="Green" title="E26">A10</font></i>: 
( <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a'</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E3:23"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="arrow.html#T2" target="_self" title="ARROW:th.2">Th2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_1_2_1"/>
( <font color="Maroon" title="c8">d</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">d</font> )
 </span><b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1_2_1_1"/><i><font color="Green" title="E27">A11</font></i>: 
<font color="Maroon" title="c8">d</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c8">d</font>
;<br/><b>take </b>
<font color="Maroon" title="c7">c'</font>
;<br/><b>thus </b><a NAME="E2:23_1_1_1_2_1_1"/>
( <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c'</font> &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c'</font> &amp; <font color="Maroon" title="c8">d</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c7">c'</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E7:23_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="arrow.html#E1:23_1_1_1_2"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="arrow.html#E3:23_1_1_1_2"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="arrow.html#E1:23_1_1_1_2_1_1"><i><font color="Green" title="E27">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_1_2_1_2"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">d</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c6">a'</font>
;<br/><b>take </b>
<font color="Maroon" title="c8">d</font>
;<br/><b>thus </b><a NAME="E2:23_1_1_1_2_1_2"/>
( <font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">d</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E5:23_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="arrow.html#E3:23_1_1_1_2"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="arrow.html#E1:23_1_1_1_2_1_2"><i><font color="Green" title="E27">A12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E10:23_1"/><i><font color="Green" title="E25">A13</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c8">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">c</font> )
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>] <b>means </b>( ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c5">b</font> ) &amp; ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c5">b</font> ) &amp; ( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c8">a</font> implies <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c8">a</font> ) &amp; ( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c8">a</font> implies <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c8">a</font> ) &amp; ( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c9">c</font> implies <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c9">c</font> ) &amp; ( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c9">c</font> implies <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c9">c</font> ) &amp; ( <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c5">b</font> ) &amp; ( <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c5">b</font> ) &amp; <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c8">a</font> );<br/>
<a NAME="E11:23_1"/><i><font color="Green" title="E26">A14</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b2">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">o</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_2_1"/>
( for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">c</font> or for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c5">b</font> )
 </span><b>by </b><i><a class="txt" href="arrow.html#E3:23_1"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_2_1_1"/>
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">c</font>
 ;<br/><div class="add"><a NAME="E2:23_1_2_1_1"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">c</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_1"><i><font color="Green" title="E25">A13</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E4:23_1_2_1_1"/><i><font color="Green" title="E28">A16</font></i>: 
( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c8">a</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="arrow.html#T7" target="_self" title="ARROW:th.7">Th7</a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">o</font>
;<br/><b>thus </b><a NAME="E5:23_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">o</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_1_2_1_1"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="arrow.html#E4:23_1_2_1_1"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a>, <a class="ref" href="arrow.html#T5" target="_self" title="ARROW:th.5">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_2_1_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c5">b</font>
 ;<br/><div class="add"><a NAME="E2:23_1_2_1_2"/><b>then </b><i><font color="Green" title="E27">A17</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_1"><i><font color="Green" title="E25">A13</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E4:23_1_2_1_2"/><i><font color="Green" title="E28">A18</font></i>: 
( <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="arrow.html#T7" target="_self" title="ARROW:th.7">Th7</a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">o</font>
;<br/><b>thus </b><a NAME="E5:23_1_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">o</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_1_2_1_2"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="arrow.html#E4:23_1_2_1_2"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a>, <a class="ref" href="arrow.html#T5" target="_self" title="ARROW:th.5">Th5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">p'</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">N</font>, <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E13:23_1"/><i><font color="Green" title="E27">A19</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">p'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="arrow.html#E11:23_1"><i><font color="Green" title="E26">A14</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c11">p'</font> = <font color="Maroon" title="c10">p'</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E15:23_1"/>
( <font color="Maroon" title="c8">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">p'</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">p'</font>,<font color="Maroon" title="c9">c</font> )
 
<b>by </b><i><a class="txt" href="arrow.html#E2:23"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="arrow.html#E10:23_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="arrow.html#E13:23_1"><i><font color="Green" title="E27">A19</font></i></a></i>;<br/>
<a NAME="E16:23_1"/><b>then </b>
( <font color="Maroon" title="c8">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">p'</font>,<font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">p'</font>,<font color="Maroon" title="c8">a</font> )
 
<b>by </b><i><a class="txt" href="arrow.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="arrow.html#E13:23_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="ref" href="arrow.html#T5" target="_self" title="ARROW:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E17:23_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:23"/><i><font color="Green" title="E22">A20</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  ex <font color="Olive" title="b2">nb</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> ex <font color="Olive" title="b3">pI</font>, <font color="Olive" title="b4">pII</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span> st <br/>( ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">nb</font> holds <br/><font color="Olive" title="b3">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font> ) &amp; ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b1">b</font>] ) &amp; ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b4">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b1">b</font>] ) &amp; ( for <font color="Olive" title="b5">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b3">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">nb</font>,<font color="Olive" title="b5">a</font> ) &amp; ( for <font color="Olive" title="b5">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b5">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b4">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">nb</font>,<font color="Olive" title="b1">b</font> ) &amp; ( for <font color="Olive" title="b5">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">pI</font>,<font color="Olive" title="b5">a</font> ) &amp; ( for <font color="Olive" title="b5">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b5">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">pII</font>,<font color="Olive" title="b1">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c4">t</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E2:23_2"/><i><font color="Green" title="E23">A21</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c4">t</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T73" title="FINSEQ_4:th.73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c4">t</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="txt" href="arrow.html#E2:23_2"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>consider </b><font color="Maroon" title="c7">oI</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E5:23_2"/><i><font color="Green" title="E24">A22</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">oI</font>,<font color="Olive" title="b1">a</font>
 <b>by </b><i><a class="ref" href="arrow.html#T8" target="_self" title="ARROW:th.8">Th8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">oII</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E7:23_2"/><i><font color="Green" title="E25">A23</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">oII</font>,<font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="arrow.html#T9" target="_self" title="ARROW:th.9">Th9</a></i>;<br/>
<a NAME="E8:23_2"/><i><font color="Green" title="E26">A24</font></i>: 
for <font color="Olive" title="b1">k0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b2">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span> st <br/>( ( for <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k0</font> holds <br/><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; ( for <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k0</font> holds <br/><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k0</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>] <b>means </b> ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> &amp; ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) );<br/>
<a NAME="E1:23_2_1"/><i><font color="Green" title="E27">A25</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b2">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">o</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<b>consider </b><font color="Maroon" title="c11">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:23_2_1_1"/><i><font color="Green" title="E28">A26</font></i>: 
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_2"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">k</font> = <font color="Maroon" title="c11">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="arrow.html#E2:23_2_1_1"><i><font color="Green" title="E28">A26</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_2_1_1_1"/>
( <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font> or <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_2_1_1_1_1"/><i><font color="Green" title="E29">A27</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c8">oII</font>
;<br/><b>thus </b><a NAME="E2:23_2_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c8">oII</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_2_1_1"><i><font color="Green" title="E28">A26</font></i></a>, <a class="txt" href="arrow.html#E1:23_2_1_1_1_1"><i><font color="Green" title="E29">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_2_1_1_1_2"/><i><font color="Green" title="E29">A28</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c7">oI</font>
;<br/><b>thus </b><a NAME="E2:23_2_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c7">oI</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_2_1_1"><i><font color="Green" title="E28">A26</font></i></a>, <a class="txt" href="arrow.html#E1:23_2_1_1_1_2"><i><font color="Green" title="E29">A28</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">p</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">N</font>, <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:23_2_1"/><i><font color="Green" title="E28">A29</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="arrow.html#E1:23_2_1"><i><font color="Green" title="E27">A25</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c11">p</font> = <font color="Maroon" title="c10">p</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">p</font>
;<br/>

<b>thus </b><a NAME="E5:23_2_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font> holds <br/><font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:23_2_1_2"/><i><font color="Green" title="E29">A30</font></i>: 
( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">i</font> = <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<a NAME="E3:23_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c11">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>]
 
<b>by </b><i><a class="txt" href="arrow.html#E3:23_2_1"><i><font color="Green" title="E28">A29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:23_2_1_2"/>
<font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font>
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_2"><i><font color="Green" title="E23">A21</font></i></a>, <a class="txt" href="arrow.html#E1:23_2_1_2"><i><font color="Green" title="E29">A30</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E6:23_2_1"/><i><font color="Green" title="E29">A31</font></i>: 
( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">i</font> = <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<a NAME="E8:23_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c11">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>]
 
<b>by </b><i><a class="txt" href="arrow.html#E3:23_2_1"><i><font color="Green" title="E28">A29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:23_2_1"/>
<font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font>
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_2"><i><font color="Green" title="E23">A21</font></i></a>, <a class="txt" href="arrow.html#E6:23_2_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>  st ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 holds <br/><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> $1 holds <br/><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) holds <br/>for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c6">b</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">kII'</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">t</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E10:23_2"/><i><font color="Green" title="E27">A32</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">kII'</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">p</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:23_2_2"/><i><font color="Green" title="E28">A33</font></i>: 
( ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">kII'</font> holds <br/><font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">kII'</font> holds <br/><font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) )
 ;<br/>

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E2:23_2_2"/><i><font color="Green" title="E29">A34</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/>

<a NAME="E3:23_2_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Maroon" title="c11">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c10">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<b>consider </b><font color="Maroon" title="c13">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:23_2_2_1"/><i><font color="Green" title="E30">A35</font></i>: 
( <font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_2"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">k</font> = <font color="Maroon" title="c13">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="arrow.html#E2:23_2_2_1"><i><font color="Green" title="E30">A35</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:23_2_2_1"/>
<font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">t</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E2:23_2_2_1"><i><font color="Green" title="E30">A35</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:23_2_2_1"/><b>then </b>
<font color="Maroon" title="c14">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">kII'</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:23_2_2_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E1:23_2_2"><i><font color="Green" title="E28">A33</font></i></a>, <a class="txt" href="arrow.html#E2:23_2_2_1"><i><font color="Green" title="E30">A35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:23_2_2_1"/>
<font color="Maroon" title="c11">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c10">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="arrow.html#E7:23_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="arrow.html#E2:23_2_2"><i><font color="Green" title="E29">A34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:23_2_2"/>
<font color="Maroon" title="c11">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="arrow.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:23_2"/><b>then </b><i><font color="Green" title="E28">A36</font></i>: 
 ex <font color="Olive" title="b1">kII'</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">kII'</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c10">kII</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E13:23_2"/><i><font color="Green" title="E29">A37</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">kII</font>] &amp; ( for <font color="Olive" title="b1">k0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k0</font>] holds <br/><font color="Olive" title="b1">k0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">kII</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="arrow.html#E11:23_2"><i><font color="Green" title="E28">A36</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c11">pII</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><br/><a NAME="E15:23_2"/><i><font color="Green" title="E30">A38</font></i>: 
( ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <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">kII</font> holds <br/><font color="Maroon" title="c11">pII</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">kII</font> holds <br/><font color="Maroon" title="c11">pII</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) )
 <b>by </b><i><a class="txt" href="arrow.html#E8:23_2"><i><font color="Green" title="E26">A24</font></i></a></i>;<br/>
<a NAME="E16:23_2"/><i><font color="Green" title="E31">A39</font></i>: 
<font color="Maroon" title="c10">kII</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_2_3"/><i><font color="Green" title="E32">A40</font></i>: 
<font color="Maroon" title="c10">kII</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>

<a NAME="E2:23_2_3"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="txt" href="arrow.html#E3:23"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E4:23_2_3"/><i><font color="Green" title="E33">A41</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="arrow.html#T1" target="_self" title="ARROW:th.1">Th1</a></i>;<br/>
<a NAME="E5:23_2_3"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c12">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<b>consider </b><font color="Maroon" title="c14">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:23_2_3_1"/><i><font color="Green" title="E34">A42</font></i>: 
( <font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_2"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">k</font> = <font color="Maroon" title="c14">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="arrow.html#E2:23_2_3_1"><i><font color="Green" title="E34">A42</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:23_2_3_1"/>
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="arrow.html#E2:23_2_3_1"><i><font color="Green" title="E34">A42</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:23_2_3_1"/><b>then </b>
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">kII</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E1:23_2_3"><i><font color="Green" title="E32">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:23_2_3_1"/><b>then </b>
<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E15:23_2"><i><font color="Green" title="E30">A38</font></i></a>, <a class="txt" href="arrow.html#E2:23_2_3_1"><i><font color="Green" title="E34">A42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:23_2_3_1"/>
<font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c12">a</font>
 <b>by </b><i><a class="txt" href="arrow.html#E5:23_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="arrow.html#E4:23_2_3"><i><font color="Green" title="E33">A41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:23_2_3"/><b>then </b>
( <font color="Maroon" title="c12">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c12">a</font> )
 
<b>by </b><i><a class="txt" href="arrow.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="arrow.html#E13:23_2"><i><font color="Green" title="E29">A37</font></i></a>, <a class="txt" href="arrow.html#E15:23_2"><i><font color="Green" title="E30">A38</font></i></a>, <a class="txt" href="arrow.html#E4:23_2_3"><i><font color="Green" title="E33">A41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:23_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">kI</font> = <font color="Maroon" title="c10">kII</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">kI</font> = <font color="Maroon" title="c12">kI</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E19:23_2"/><i><font color="Green" title="E32">A43</font></i>: 
<font color="Maroon" title="c10">kII</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">kI</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
;<br/>
<a NAME="E20:23_2"/>
<font color="Maroon" title="c13">kI</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="arrow.html#E16:23_2"><i><font color="Green" title="E31">A39</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E21:23_2"/><b>then </b><i><font color="Green" title="E33">A44</font></i>: 
<font color="Maroon" title="c13">kI</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/>
<a NAME="E22:23_2"/>
<font color="Maroon" title="c10">kII</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">kII'</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E10:23_2"><i><font color="Green" title="E27">A32</font></i></a>, <a class="txt" href="arrow.html#E13:23_2"><i><font color="Green" title="E29">A37</font></i></a></i>;<br/>
<a NAME="E23:23_2"/><b>then </b>
<font color="Maroon" title="c13">kI</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">kII'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E24:23_2"/><b>then </b><i><font color="Green" title="E34">A45</font></i>: 
<font color="Maroon" title="c13">kI</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E21:23_2"><i><font color="Green" title="E33">A44</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">nb</font> = <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">kI</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<a NAME="E26:23_2"/><i><font color="Green" title="E35">A46</font></i>: 
<font color="Maroon" title="c13">kI</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">kII</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E19:23_2"><i><font color="Green" title="E32">A43</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">pI</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><br/><a NAME="E28:23_2"/><i><font color="Green" title="E36">A47</font></i>: 
( ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">kI</font> holds <br/><font color="Maroon" title="c15">pI</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">kI</font> holds <br/><font color="Maroon" title="c15">pI</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) &amp;  ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; not <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pI</font>,<font color="Maroon" title="c6">b</font> ) )
 <b>by </b><i><a class="txt" href="arrow.html#E13:23_2"><i><font color="Green" title="E29">A37</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">nb</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c15">pI</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c11">pII</font>
;<br/>

<b>thus </b><a NAME="E29:23_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">nb</font> holds <br/><font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<b>assume </b><a NAME="E1:23_2_4"/><i><font color="Green" title="E37">A48</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">nb</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c17">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:23_2_4"/><i><font color="Green" title="E38">A49</font></i>: 
( <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_2"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c17">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="arrow.html#E3:23_2_4"><i><font color="Green" title="E38">A49</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_2_4_1"/>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">kI</font> or <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">kI</font> )
 </span><b>by </b><i><a class="txt" href="arrow.html#E1:23_2_4"><i><font color="Green" title="E37">A48</font></i></a>, <a class="txt" href="arrow.html#E3:23_2_4"><i><font color="Green" title="E38">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_4_1_1"><b>suppose </b></a><a NAME="E1:23_2_4_1_1"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">kI</font>
 ;<br/><div class="add"><a NAME="E2:23_2_4_1_1"/><b>then </b>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">kI</font> &amp; <font color="Maroon" title="c18">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">kII</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E19:23_2"><i><font color="Green" title="E32">A43</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E3:23_2_4_1_1"/><b>then </b>
( <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> &amp; <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E15:23_2"><i><font color="Green" title="E30">A38</font></i></a>, <a class="txt" href="arrow.html#E28:23_2"><i><font color="Green" title="E36">A47</font></i></a>, <a class="txt" href="arrow.html#E3:23_2_4"><i><font color="Green" title="E38">A49</font></i></a></i>;<br/><b>hence </b><a NAME="E4:23_2_4_1_1"/>
<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_4_1_2"><b>suppose </b></a><a NAME="E1:23_2_4_1_2"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">kI</font>
 ;<br/><div class="add"><a NAME="E2:23_2_4_1_2"/><b>then </b>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">kI</font> &amp; <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">kII</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E19:23_2"><i><font color="Green" title="E32">A43</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E3:23_2_4_1_2"/><b>then </b>
( <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> &amp; <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E15:23_2"><i><font color="Green" title="E30">A38</font></i></a>, <a class="txt" href="arrow.html#E28:23_2"><i><font color="Green" title="E36">A47</font></i></a>, <a class="txt" href="arrow.html#E3:23_2_4"><i><font color="Green" title="E38">A49</font></i></a></i>;<br/><b>hence </b><a NAME="E4:23_2_4_1_2"/>
<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E30:23_2"/><i><font color="Green" title="E37">A50</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">b</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<b>consider </b><font color="Maroon" title="c17">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:23_2_5"/><i><font color="Green" title="E38">A51</font></i>: 
( <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_2"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c17">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="arrow.html#E2:23_2_5"><i><font color="Green" title="E38">A51</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:23_2_5"/>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">kI</font> or <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">kI</font> )
 
;<br/>
<a NAME="E5:23_2_5"/><b>then </b>
( <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> or <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> )
 
<b>by </b><i><a class="txt" href="arrow.html#E28:23_2"><i><font color="Green" title="E36">A47</font></i></a>, <a class="txt" href="arrow.html#E2:23_2_5"><i><font color="Green" title="E38">A51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:23_2_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c6">b</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E5:23_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="arrow.html#E7:23_2"><i><font color="Green" title="E25">A23</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E31:23_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">b</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<b>consider </b><font color="Maroon" title="c17">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:23_2_6"/><i><font color="Green" title="E38">A52</font></i>: 
( <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_2"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c17">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="arrow.html#E2:23_2_6"><i><font color="Green" title="E38">A52</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:23_2_6"/>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">kII</font> or <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">kII</font> )
 
;<br/>
<a NAME="E5:23_2_6"/><b>then </b>
( <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> or <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> )
 
<b>by </b><i><a class="txt" href="arrow.html#E15:23_2"><i><font color="Green" title="E30">A38</font></i></a>, <a class="txt" href="arrow.html#E2:23_2_6"><i><font color="Green" title="E38">A52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:23_2_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c6">b</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E5:23_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="arrow.html#E7:23_2"><i><font color="Green" title="E25">A23</font></i></a></i>;<br/>


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

<a NAME="E32:23_2"/>
<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E24:23_2"><i><font color="Green" title="E34">A45</font></i></a>, <a class="txt" href="arrow.html#E28:23_2"><i><font color="Green" title="E36">A47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E33:23_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Olive" title="b1">a</font>
 <b>by </b><i><a class="txt" href="arrow.html#E5:23_2"><i><font color="Green" title="E24">A22</font></i></a></i>;<br/>

<a NAME="E34:23_2"/>
<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E15:23_2"><i><font color="Green" title="E30">A38</font></i></a>, <a class="txt" href="arrow.html#E24:23_2"><i><font color="Green" title="E34">A45</font></i></a>, <a class="txt" href="arrow.html#E26:23_2"><i><font color="Green" title="E35">A46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E35:23_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="arrow.html#E7:23_2"><i><font color="Green" title="E25">A23</font></i></a></i>;<br/>

<b>thus </b><a NAME="E36:23_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pI</font>,<font color="Olive" title="b1">a</font>
 <b>by </b><i><a class="txt" href="arrow.html#E4:23"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="arrow.html#E28:23_2"><i><font color="Green" title="E36">A47</font></i></a>, <a class="txt" href="arrow.html#E30:23_2"><i><font color="Green" title="E37">A50</font></i></a></i>;<br/>

<b>thus </b><a NAME="E37:23_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="arrow.html#E13:23_2"><i><font color="Green" title="E29">A37</font></i></a>, <a class="txt" href="arrow.html#E15:23_2"><i><font color="Green" title="E30">A38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:23"/><i><font color="Green" title="E23">A53</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  ex <font color="Olive" title="b2">nb</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> ex <font color="Olive" title="b3">pI</font>, <font color="Olive" title="b4">pII</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span> st <br/>( ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">nb</font> holds <br/><font color="Olive" title="b3">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font> ) &amp; ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b1">b</font>] ) &amp; ( for <font color="Olive" title="b5">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">pI</font>,<font color="Olive" title="b5">a</font> ) &amp; ( for <font color="Olive" title="b5">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b5">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">pII</font>,<font color="Olive" title="b1">b</font> ) &amp; ( for <font color="Olive" title="b5">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b6">a</font>, <font color="Olive" title="b7">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b6">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">nb</font>,<font color="Olive" title="b7">c</font> holds <br/><font color="Olive" title="b6">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">p</font>,<font color="Olive" title="b7">c</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>consider </b><font color="Maroon" title="c5">nb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>, <font color="Maroon" title="c6">pI</font>, <font color="Maroon" title="c7">pII</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><br/><a NAME="E2:23_3"/><i><font color="Green" title="E24">A54</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> holds <br/><font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>
 <b>and </b><br/><a NAME="E3:23_3"/><i><font color="Green" title="E25">A55</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font>]
 <b>and </b><br/><a NAME="E4:23_3"/><i><font color="Green" title="E26">A56</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font>]
 <b>and </b><br/><a NAME="E5:23_3"/><i><font color="Green" title="E27">A57</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Olive" title="b1">a</font>
 <b>and </b><br/><a NAME="E6:23_3"/><i><font color="Green" title="E28">A58</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Maroon" title="c4">b</font>
 <b>and </b><br/><a NAME="E7:23_3"/><i><font color="Green" title="E29">A59</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Olive" title="b1">a</font>
 <b>and </b><br/><a NAME="E8:23_3"/><i><font color="Green" title="E30">A60</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="arrow.html#E5:23"><i><font color="Green" title="E22">A20</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">nb</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">pI</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">pII</font>
;<br/>

<b>thus </b><a NAME="E9:23_3"/>
( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> holds <br/><font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font>] ) &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Olive" title="b1">a</font> ) &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font> ) )
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_3"><i><font color="Green" title="E24">A54</font></i></a>, <a class="txt" href="arrow.html#E3:23_3"><i><font color="Green" title="E25">A55</font></i></a>, <a class="txt" href="arrow.html#E7:23_3"><i><font color="Green" title="E29">A59</font></i></a>, <a class="txt" href="arrow.html#E8:23_3"><i><font color="Green" title="E30">A60</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c8">p</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E10:23_3"/><i><font color="Green" title="E31">A61</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> )
 <b>and </b><br/><a NAME="E11:23_3"/><i><font color="Green" title="E32">A62</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Maroon" title="c10">c</font>
 ;<br/>

<a NAME="E12:23_3"/><i><font color="Green" title="E33">A63</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E11:23_3"><i><font color="Green" title="E32">A62</font></i></a>, <a class="ref" href="arrow.html#T3" target="_self" title="ARROW:th.3">Th3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>] <b>means </b>( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c9">a</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font> implies ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c10">c</font> ) ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> implies ( ( ( for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Olive" title="b1">d</font> ) implies ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c10">c</font> ) ) &amp; ( ( for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c4">b</font> ) implies ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c4">b</font> ) ) ) ) );<br/>
<a NAME="E13:23_3"/><i><font color="Green" title="E34">A64</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b2">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">o</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_3_1_1"/>
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font> or <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_1"><b>suppose </b></a><a NAME="E1:23_3_1_1_1"/><i><font color="Green" title="E35">A65</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c12">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:23_3_1_1_1"/><i><font color="Green" title="E36">A66</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="txt" href="arrow.html#E12:23_3"><i><font color="Green" title="E33">A63</font></i></a>, <a class="ref" href="arrow.html#T7" target="_self" title="ARROW:th.7">Th7</a></i>;<br/><b>take </b>
<font color="Maroon" title="c12">o</font>
;<br/><b>thus </b><a NAME="E4:23_3_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">o</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E11:23_3"><i><font color="Green" title="E32">A62</font></i></a>, <a class="txt" href="arrow.html#E1:23_3_1_1_1"><i><font color="Green" title="E35">A65</font></i></a>, <a class="txt" href="arrow.html#E3:23_3_1_1_1"><i><font color="Green" title="E36">A66</font></i></a>, <a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a>, <a class="ref" href="arrow.html#T5" target="_self" title="ARROW:th.5">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_2"><b>suppose </b></a><a NAME="E1:23_3_1_1_2"/><i><font color="Green" title="E35">A67</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_3_1_1_2_1"/>
( for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">d</font> or for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c4">b</font> )
 </span><b>by </b><i><a class="txt" href="arrow.html#E4:23_3"><i><font color="Green" title="E26">A56</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:23_3_1_1_2_1_1"/>
for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">d</font>
 ;<br/><div class="add"><a NAME="E2:23_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a></i>;<br/><a NAME="E3:23_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E36">A68</font></i>: 
not <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E5:23_3_1_1_2_1_1"/><i><font color="Green" title="E37">A69</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c9">a</font> ) )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="ref" href="arrow.html#T10" target="_self" title="ARROW:th.10">Th10</a></i>;<br/><b>take </b>
<font color="Maroon" title="c12">o</font>
;<br/><b>thus </b><a NAME="E6:23_3_1_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">o</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="txt" href="arrow.html#E1:23_3_1_1_2"><i><font color="Green" title="E35">A67</font></i></a>, <a class="txt" href="arrow.html#E3:23_3_1_1_2_1_1"><i><font color="Green" title="E36">A68</font></i></a>, <a class="txt" href="arrow.html#E5:23_3_1_1_2_1_1"><i><font color="Green" title="E37">A69</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:23_3_1_1_2_1_2"/>
for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c4">b</font>
 ;<br/><div class="add"><a NAME="E2:23_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a></i>;<br/><a NAME="E3:23_3_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E36">A70</font></i>: 
not <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E5:23_3_1_1_2_1_2"/><i><font color="Green" title="E37">A71</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c4">b</font> &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c9">a</font> ) )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="ref" href="arrow.html#T11" target="_self" title="ARROW:th.11">Th11</a></i>;<br/><b>take </b>
<font color="Maroon" title="c12">o</font>
;<br/><b>thus </b><a NAME="E6:23_3_1_1_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">o</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="txt" href="arrow.html#E1:23_3_1_1_2"><i><font color="Green" title="E35">A67</font></i></a>, <a class="txt" href="arrow.html#E3:23_3_1_1_2_1_2"><i><font color="Green" title="E36">A70</font></i></a>, <a class="txt" href="arrow.html#E5:23_3_1_1_2_1_2"><i><font color="Green" title="E37">A71</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c11">pIII</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">N</font>, <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E15:23_3"/><i><font color="Green" title="E35">A72</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c11">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="arrow.html#E13:23_3"><i><font color="Green" title="E34">A64</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c12">pIII</font> = <font color="Maroon" title="c11">pIII</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E17:23_3"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_3_2_1"/>
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font> or <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_2_1_1"><b>suppose </b></a><a NAME="E1:23_3_2_1_1"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font>
 ;<br/><div class="add"><a NAME="E2:23_3_2_1_1"/><b>then </b>
( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E6:23_3"><i><font color="Green" title="E28">A58</font></i></a>, <a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="txt" href="arrow.html#E15:23_3"><i><font color="Green" title="E35">A72</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_3_2_1_1"/>
( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) )
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_2_1_2"><b>suppose </b></a><a NAME="E1:23_3_2_1_2"/><i><font color="Green" title="E36">A73</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_3_2_1_2_1"/>
( for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Olive" title="b1">d</font> or for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> )
 </span><b>by </b><i><a class="txt" href="arrow.html#E4:23_3"><i><font color="Green" title="E26">A56</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:23_3_2_1_2_1_1"/>
for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Olive" title="b1">d</font>
 ;<br/><div class="add"><a NAME="E2:23_3_2_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="txt" href="arrow.html#E15:23_3"><i><font color="Green" title="E35">A72</font></i></a>, <a class="txt" href="arrow.html#E1:23_3_2_1_2"><i><font color="Green" title="E36">A73</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_3_2_1_2_1_1"/>
( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) )
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:23_3_2_1_2_1_2"/>
for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font>
 ;<br/><div class="add"><a NAME="E2:23_3_2_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="txt" href="arrow.html#E15:23_3"><i><font color="Green" title="E35">A72</font></i></a>, <a class="txt" href="arrow.html#E1:23_3_2_1_2"><i><font color="Green" title="E36">A73</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_3_2_1_2_1_2"/>
( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) )
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E18:23_3"/><b>then </b><i><font color="Green" title="E36">A74</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font> iff <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">pIII</font>,<font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="arrow.html#E2:23"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<a NAME="E19:23_3"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_3_3_1"/>
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font> or <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_3_1_1"><b>suppose </b></a><a NAME="E1:23_3_3_1_1"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font>
 ;<br/><div class="add"><a NAME="E2:23_3_3_1_1"/><b>then </b>
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E5:23_3"><i><font color="Green" title="E27">A57</font></i></a>, <a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="txt" href="arrow.html#E15:23_3"><i><font color="Green" title="E35">A72</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_3_3_1_1"/>
( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) )
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_3_1_2"><b>suppose </b></a><a NAME="E1:23_3_3_1_2"/><i><font color="Green" title="E37">A75</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_3_3_1_2_1"/>
( for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Olive" title="b1">d</font> or for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> )
 </span><b>by </b><i><a class="txt" href="arrow.html#E4:23_3"><i><font color="Green" title="E26">A56</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:23_3_3_1_2_1_1"/>
for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Olive" title="b1">d</font>
 ;<br/><div class="add"><a NAME="E2:23_3_3_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="txt" href="arrow.html#E15:23_3"><i><font color="Green" title="E35">A72</font></i></a>, <a class="txt" href="arrow.html#E1:23_3_3_1_2"><i><font color="Green" title="E37">A75</font></i></a></i>;<br/><a NAME="E3:23_3_3_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_3"><i><font color="Green" title="E24">A54</font></i></a>, <a class="txt" href="arrow.html#E1:23_3_3_1_2"><i><font color="Green" title="E37">A75</font></i></a></i>;<br/><b>hence </b><a NAME="E4:23_3_3_1_2_1_1"/>
( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) )
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:23_3_3_1_2_1_2"/>
for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font>
 ;<br/><div class="add"><a NAME="E2:23_3_3_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="txt" href="arrow.html#E15:23_3"><i><font color="Green" title="E35">A72</font></i></a>, <a class="txt" href="arrow.html#E1:23_3_3_1_2"><i><font color="Green" title="E37">A75</font></i></a></i>;<br/><a NAME="E3:23_3_3_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:23_3"><i><font color="Green" title="E24">A54</font></i></a>, <a class="txt" href="arrow.html#E1:23_3_3_1_2"><i><font color="Green" title="E37">A75</font></i></a></i>;<br/><b>hence </b><a NAME="E4:23_3_3_1_2_1_2"/>
( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) )
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E20:23_3"/><b>then </b>
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Maroon" title="c10">c</font> iff <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">pIII</font>,<font color="Maroon" title="c10">c</font> )
 
<b>by </b><i><a class="txt" href="arrow.html#E2:23"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<a NAME="E21:23_3"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">pIII</font>,<font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="txt" href="arrow.html#E7:23_3"><i><font color="Green" title="E29">A59</font></i></a>, <a class="txt" href="arrow.html#E8:23_3"><i><font color="Green" title="E30">A60</font></i></a>, <a class="txt" href="arrow.html#E10:23_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="txt" href="arrow.html#E18:23_3"><i><font color="Green" title="E36">A74</font></i></a>, <a class="ref" href="arrow.html#T5" target="_self" title="ARROW:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E22:23_3"/>
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="txt" href="arrow.html#E2:23"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="arrow.html#E15:23_3"><i><font color="Green" title="E35">A72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">nb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>, <font color="Maroon" title="c6">pI</font>, <font color="Maroon" title="c7">pII</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><br/><a NAME="E9:23"/><i><font color="Green" title="E24">A76</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> holds <br/><font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>
 <b>and </b><br/><a NAME="E10:23"/><i><font color="Green" title="E25">A77</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font>]
 <b>and </b><br/><a NAME="E11:23"/><i><font color="Green" title="E26">A78</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Olive" title="b1">a</font>
 <b>and </b><br/><a NAME="E12:23"/><i><font color="Green" title="E27">A79</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font>
 <b>and </b><br/><a NAME="E13:23"/><i><font color="Green" title="E28">A80</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">c</font>
 <b>by </b><i><a class="txt" href="arrow.html#E6:23"><i><font color="Green" title="E23">A53</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">nb</font>
;<br/>

<b>let </b><font color="Maroon" title="c8">p</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">a'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>





<b>assume </b><a NAME="E14:23"/><i><font color="Green" title="E29">A81</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Maroon" title="c10">a'</font>
 ;<br/>

<a NAME="E15:23"/><b>then </b><i><font color="Green" title="E30">A82</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a'</font>
 
<b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_4"/>
( ( <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> ) or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c10">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1"><b>suppose </b></a><a NAME="E1:23_4_1"/>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_4_1"/>
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">a'</font>
 <b>by </b><i><a class="txt" href="arrow.html#E13:23"><i><font color="Green" title="E28">A80</font></i></a>, <a class="txt" href="arrow.html#E14:23"><i><font color="Green" title="E29">A81</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_2"><b>suppose </b></a><a NAME="E1:23_4_2"/><i><font color="Green" title="E31">A83</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c10">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c11">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:23_4_2"/><i><font color="Green" title="E32">A84</font></i>: 
( <font color="Maroon" title="c11">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c11">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a'</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E3:23"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="arrow.html#T2" target="_self" title="ARROW:th.2">Th2</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">nc</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>, <font color="Maroon" title="c13">pI'</font>, <font color="Maroon" title="c14">pII'</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><br/><a NAME="E5:23_4_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">nc</font> holds <br/><font color="Maroon" title="c13">pI'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">pII'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>
 <b>and </b><br/><a NAME="E6:23_4_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">pI'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c11">c</font>]
 <b>and </b><br/><a NAME="E7:23_4_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c</font> holds <br/><font color="Maroon" title="c11">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">pI'</font>,<font color="Olive" title="b1">a</font>
 <b>and </b><br/><a NAME="E8:23_4_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">pII'</font>,<font color="Maroon" title="c11">c</font>
 <b>and </b><br/><a NAME="E9:23_4_2"/><i><font color="Green" title="E33">A85</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">a'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c</font> &amp; <font color="Olive" title="b3">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c</font> &amp; <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Olive" title="b3">a'</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">a'</font>
 <b>by </b><i><a class="txt" href="arrow.html#E6:23"><i><font color="Green" title="E23">A53</font></i></a></i>;<br/><a NAME="E10:23_4_2"/>
<font color="Maroon" title="c12">nc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_4_2_1_1"/>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c10">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 </span><b>by </b><i><a class="txt" href="arrow.html#E1:23_4_2"><i><font color="Green" title="E31">A83</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_2_1_1_1"><b>suppose </b></a><a NAME="E1:23_4_2_1_1_1"/><i><font color="Green" title="E34">A86</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:23_4_2_1_1_1"/><i><font color="Green" title="E35">A87</font></i>: 
<font color="Maroon" title="c12">nc</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font>
 ;<br/><a NAME="E3:23_4_2_1_1_1"/>
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c10">a'</font> or <font color="Maroon" title="c10">a'</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23"><i><font color="Green" title="E25">A77</font></i></a>, <a class="txt" href="arrow.html#E15:23"><i><font color="Green" title="E30">A82</font></i></a>, <a class="txt" href="arrow.html#E1:23_4_2_1_1_1"><i><font color="Green" title="E34">A86</font></i></a></i>;<br/><a NAME="E4:23_4_2_1_1_1"/><b>then </b>
( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c10">a'</font> &amp; <font color="Maroon" title="c10">a'</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font> ) or ( <font color="Maroon" title="c10">a'</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Maroon" title="c10">a'</font> ) )
 <b>by </b><i><a class="txt" href="arrow.html#E9:23"><i><font color="Green" title="E24">A76</font></i></a>, <a class="txt" href="arrow.html#E11:23"><i><font color="Green" title="E26">A78</font></i></a>, <a class="txt" href="arrow.html#E12:23"><i><font color="Green" title="E27">A79</font></i></a>, <a class="txt" href="arrow.html#E15:23"><i><font color="Green" title="E30">A82</font></i></a>, <a class="txt" href="arrow.html#E1:23_4_2_1_1_1"><i><font color="Green" title="E34">A86</font></i></a>, <a class="txt" href="arrow.html#E2:23_4_2_1_1_1"><i><font color="Green" title="E35">A87</font></i></a></i>;<br/><a NAME="E5:23_4_2_1_1_1"/><b>then </b>
( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c10">a'</font> &amp; <font color="Maroon" title="c10">a'</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font> ) or ( <font color="Maroon" title="c10">a'</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Maroon" title="c10">a'</font> ) )
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E6:23_4_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arrow.html#E3:23_4_2"><i><font color="Green" title="E32">A84</font></i></a>, <a class="txt" href="arrow.html#E9:23_4_2"><i><font color="Green" title="E33">A85</font></i></a>, <a class="txt" href="arrow.html#E1:23_4_2_1_1_1"><i><font color="Green" title="E34">A86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_2_1_1_2"><b>suppose </b></a><a NAME="E1:23_4_2_1_1_2"/><i><font color="Green" title="E34">A88</font></i>: 
<font color="Maroon" title="c10">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:23_4_2_1_1_2"/><i><font color="Green" title="E35">A89</font></i>: 
<font color="Maroon" title="c12">nc</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font>
 ;<br/><a NAME="E3:23_4_2_1_1_2"/>
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c9">a</font> or <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:23"><i><font color="Green" title="E25">A77</font></i></a>, <a class="txt" href="arrow.html#E15:23"><i><font color="Green" title="E30">A82</font></i></a>, <a class="txt" href="arrow.html#E1:23_4_2_1_1_2"><i><font color="Green" title="E34">A88</font></i></a></i>;<br/><a NAME="E4:23_4_2_1_1_2"/><b>then </b>
( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font> ) or ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Maroon" title="c9">a</font> ) )
 <b>by </b><i><a class="txt" href="arrow.html#E9:23"><i><font color="Green" title="E24">A76</font></i></a>, <a class="txt" href="arrow.html#E11:23"><i><font color="Green" title="E26">A78</font></i></a>, <a class="txt" href="arrow.html#E12:23"><i><font color="Green" title="E27">A79</font></i></a>, <a class="txt" href="arrow.html#E15:23"><i><font color="Green" title="E30">A82</font></i></a>, <a class="txt" href="arrow.html#E1:23_4_2_1_1_2"><i><font color="Green" title="E34">A88</font></i></a>, <a class="txt" href="arrow.html#E2:23_4_2_1_1_2"><i><font color="Green" title="E35">A89</font></i></a></i>;<br/><a NAME="E5:23_4_2_1_1_2"/><b>then </b>
( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c9">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font> ) or ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Maroon" title="c9">a</font> ) )
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E6:23_4_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arrow.html#E3:23_4_2"><i><font color="Green" title="E32">A84</font></i></a>, <a class="txt" href="arrow.html#E9:23_4_2"><i><font color="Green" title="E33">A85</font></i></a>, <a class="txt" href="arrow.html#E1:23_4_2_1_1_2"><i><font color="Green" title="E34">A88</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E11:23_4_2"/>
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">a'</font>
 <b>by </b><i><a class="txt" href="arrow.html#E14:23"><i><font color="Green" title="E29">A81</font></i></a>, <a class="txt" href="arrow.html#E3:23_4_2"><i><font color="Green" title="E32">A84</font></i></a>, <a class="txt" href="arrow.html#E9:23_4_2"><i><font color="Green" title="E33">A85</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
