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

<b>consider </b><font color="Maroon" title="c4">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E2:20_1_1"/><i><font color="Green" title="E15">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T73" title="FINSEQ_1:th.73">FINSEQ_1:73</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <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="c4">p</font> implies  ex <font color="Olive" title="b1">a</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ex <font color="Olive" title="b2">A</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">a</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="c4">p</font> &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> &amp; ( for <font color="Olive" title="b3">c</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b4">C</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b3">c</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="c4">p</font> &amp; <font color="Olive" title="b3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">C</font> holds <br/><font color="Olive" title="b4">C</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b2">A</font>,<font color="Maroon" title="c2">T</font> ) ) );<br/>
<a NAME="E3:20_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="lexbfs.html#E2:20_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="finseq_1.html#T27" title="FINSEQ_1:th.27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E4:20_1_1"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E5:20_1_1"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_1"/><i><font color="Green" title="E18">A4</font></i>: 
1 <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="c4">p</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E2:20_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="finseq_3.html#T34" title="FINSEQ_3:th.34">FINSEQ_3:34</a></i>;<br/>
<a NAME="E2:20_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E2:20_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">c</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c7">C</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>assume </b><a NAME="E1:20_1_1_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c6">c</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="c4">p</font> &amp; <font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C</font> )
 ;<br/><a NAME="E2:20_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:20_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">C</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_1_1"/>
<font color="Maroon" title="c7">C</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#T6" title="TERMORD:th.6">TERMORD:6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:20_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_1"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:20_1_1"/><i><font color="Green" title="E18">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_1_2"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_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="E1:20_1_1_2_1"/>
( <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> or <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>, <font color="Maroon" title="c7">A</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:20_1_1_2_1_1"/><i><font color="Green" title="E21">A9</font></i>: 
( <font color="Maroon" title="c6">a</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="c4">p</font> &amp; <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> )
 <b>and </b><br/><a NAME="E4:20_1_1_2_1_1"/><i><font color="Green" title="E22">A10</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">C</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">c</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="c4">p</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Olive" title="b2">C</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_2"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/><a NAME="E5:20_1_1_2_1_1"/><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c4">p</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_2_1_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:20_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="chord.html#T1" title="CHORD:th.1">CHORD:1</a></i>;<br/><a NAME="E7:20_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c4">p</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:20_1_1_2_1_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:20_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">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">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E2:20_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">Ck</font> = <font color="Maroon" title="c4">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">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>set </b><font color="Maroon" title="c9">m</font> =  <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">Ck</font>,<font color="Maroon" title="c2">T</font>;<br/><a NAME="E10:20_1_1_2_1_1"/><i><font color="Green" title="E25">A13</font></i>: 
( <font color="Maroon" title="c7">A</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">Ck</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c8">Ck</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">Ck</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="termord.html#T14" title="TERMORD:th.14">TERMORD:14</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_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="E1:20_1_1_2_1_1_1"/>
(  <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">Ck</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">A</font> or  <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">Ck</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">Ck</font> )
 </span><b>by </b><i><a class="ref" href="termord.html#T12" title="TERMORD:th.12">TERMORD:12</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_1_1"/><i><font color="Green" title="E26">A14</font></i>: 
 <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">Ck</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">A</font>
 ;<br/><div class="add"><a NAME="E2:20_1_1_2_1_1_1_1"/><i><font color="Green" title="E27">A15</font></i>: 
( <font color="Maroon" title="c6">a</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="c4">p</font> &amp; <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E3:20_1_1_2_1_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">c</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c11">C</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>assume </b><a NAME="E1:20_1_1_2_1_1_1_1_1"/><i><font color="Green" title="E28">A16</font></i>: 
( <font color="Maroon" title="c10">c</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="c4">p</font> &amp; <font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">C</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_2_1_1_1_1_1_1"/>
( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_2_1_1_1_1_1"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">C</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">Ck</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_2_1_1_1_1_1"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="termord.html#T14" title="TERMORD:th.14">TERMORD:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:20_1_1_2_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:20_1_1_2_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">C</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_2_1_1_1_1_1"><i><font color="Green" title="E28">A16</font></i></a>, <a class="txt" href="lexbfs.html#E4:20_1_1_2_1_1"><i><font color="Green" title="E22">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c11">C</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">Ck</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E10:20_1_1_2_1_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="termord.html#T8" title="TERMORD:th.8">TERMORD:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:20_1_1_2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_2_1_1_1_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="txt" href="lexbfs.html#E2:20_1_1_2_1_1_1_1"><i><font color="Green" title="E27">A15</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_1_2"/><i><font color="Green" title="E26">A17</font></i>: 
 <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">Ck</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">Ck</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">c</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c11">C</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>assume </b><a NAME="E1:20_1_1_2_1_1_1_2_1"/><i><font color="Green" title="E27">A18</font></i>: 
( <font color="Maroon" title="c10">c</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="c4">p</font> &amp; <font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">C</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_2_1_1_1_2_1_1"/>
( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_2_1_1_1_2_1"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">C</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">Ck</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_2_1_1_1_2_1"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="termord.html#T14" title="TERMORD:th.14">TERMORD:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:20_1_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:20_1_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">C</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_2_1_1_1_2_1"><i><font color="Green" title="E27">A18</font></i></a>, <a class="txt" href="lexbfs.html#E4:20_1_1_2_1_1"><i><font color="Green" title="E22">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c11">C</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">Ck</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E10:20_1_1_2_1_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="termord.html#T8" title="TERMORD:th.8">TERMORD:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:20_1_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:20_1_1_2_1_1_1_2"><i><font color="Green" title="E26">A17</font></i></a>, <a class="txt" href="lexbfs.html#E7:20_1_1_2_1_1"><i><font color="Green" title="E24">A12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_2"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:20_1_1"/>
for <font color="Olive" title="b1">k</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="lexbfs.html#E5:20_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="lexbfs.html#E6:20_1_1"><i><font color="Green" title="E18">A6</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c5">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>, <font color="Maroon" title="c6">A</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E9:20_1_1"/><i><font color="Green" title="E19">A19</font></i>: 
( <font color="Maroon" title="c5">a</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="c4">p</font> &amp; <font color="Maroon" title="c5">a</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="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> )
 <b>and </b><br/><a NAME="E10:20_1_1"/><i><font color="Green" title="E20">A20</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">C</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">c</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="c4">p</font> &amp; <font color="Olive" title="b1">c</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="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Olive" title="b2">C</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E4:20_1_1"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">A</font>
;<br/>

<b>thus </b><a NAME="E11:20_1_1"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E9:20_1_1"><i><font color="Green" title="E19">A19</font></i></a>, <a class="txt" href="lexbfs.html#E2:20_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>assume </b><a NAME="E1:20_1_1_3"/><i><font color="Green" title="E21">A21</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ;<br/><b>consider </b><font color="Maroon" title="c8">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:20_1_1_3"/><i><font color="Green" title="E22">A22</font></i>: 
( <font color="Maroon" title="c8">y</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="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E2:20_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="lexbfs.html#E1:20_1_1_3"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><a NAME="E4:20_1_1_3"/>
<font color="Maroon" title="c8">y</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="c4">p</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E3:20_1_1_3"><i><font color="Green" title="E22">A22</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E5:20_1_1_3"/>
<font color="Maroon" title="c7">x</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E3:20_1_1_3"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="lexbfs.html#E10:20_1_1"><i><font color="Green" title="E20">A20</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:20_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> holds <br/><font color="Olive" title="b1">x</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c2">T</font>
 ;<br/>


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