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

<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="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font></span>)</span> or  ex <font color="Olive" title="b1">M</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">b'</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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b2">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b3">b'</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b3">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> ) ) );<br/>
<a NAME="E1:24_1_1"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_1_1_1"/>
 ex <font color="Olive" title="b1">M</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">b'</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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b2">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b3">b'</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b3">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c6">M</font> =  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<b>take </b>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E1:24_1_1_1_1"/>
(  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">b'</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">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b1">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E21">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:24_1_1_2"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ;<br/><a NAME="E2:24_1_1_2"/>
( <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font></span>)</span> or  ex <font color="Olive" title="b1">M</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">b'</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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b2">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b3">b'</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b3">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_2_1"/><i><font color="Green" title="E23">A5</font></i>: 
not <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font></span>)</span>
 ;<br/>

<a NAME="E2:24_1_1_2_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">M1</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E4:24_1_1_2_1"/><i><font color="Green" title="E24">A6</font></i>: 
( <font color="Maroon" title="c7">M1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> &amp; ( for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">b'</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">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M1</font> &amp; <font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b1">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M1</font> ) )
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_1_2"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="groeb_3.html#E1:24_1_1_2_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:24_1_1_2_1"/><i><font color="Green" title="E25">A7</font></i>: 
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M1</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E4:24_1_1_2_1"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">G</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c7">M1</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_1_2_1_1"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c7">M1</font>
 ;<br/><a NAME="E2:24_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_2_1_1"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">G</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c7">M1</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_1_2_1_2"/>
<font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:24_1_1_2_1_2"/><b>then </b>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">M1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:24_1_1_2_1_2"/><b>then </b>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M1</font>
 ;<br/><a NAME="E4:24_1_1_2_1_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E4:24_1_1_2_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="groeb_3.html#E5:24_1_1_2_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E5:24_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_1_2_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="nat_1.html#T16" title="NAT_1:th.16">NAT_1:16</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">G</font> = <font color="Maroon" title="c9">G</font> as  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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c11">R</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #);<br/>
<b>consider </b><font color="Maroon" title="c12">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #)<b> such that </b><br/><a NAME="E11:24_1_1_2_1"/><i><font color="Green" title="E26">A8</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font> &amp; <font color="Maroon" title="c12">x</font> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c10">G</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #) )
 <b>by </b><i><a class="ref" href="bagorder.html#T7" title="BAGORDER:th.7">BAGORDER:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">b</font> = <font color="Maroon" title="c12">x</font> 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><i><font color="Green" title="E27">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">b'</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="E1:24_1_1_2_1_3"/><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon" title="c14">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_2_1_3_1_1"/>
( <font color="Maroon" title="c14">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> or <font color="Maroon" title="c14">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_3_1_1_1"><b>case </b></a><a NAME="E1:24_1_1_2_1_3_1_1_1"/>
<font color="Maroon" title="c14">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1_2_1_3_1_1_1"/>
<font color="Maroon" title="c14">b'</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c13">b</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_3_1_1_2"><b>case </b></a><a NAME="E1:24_1_1_2_1_3_1_1_2"/>
<font color="Maroon" title="c14">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">b</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_2_1_3_1_1_2"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">b'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E11:24_1_1_2_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="groeb_3.html#E1:24_1_1_2_1_3"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="waybel_4.html#D24" title="WAYBEL_4:def.24">WAYBEL_4:def 24</a></i>;<br/><a NAME="E3:24_1_1_2_1_3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c13">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#D2" title="TERMORD:def.2">TERMORD:def 2</a></i>;<br/><a NAME="E4:24_1_1_2_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">b'</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#T5" title="TERMORD:th.5">TERMORD:5</a></i>;<br/><b>hence </b><a NAME="E5:24_1_1_2_1_3_1_1_2"/>
<font color="Maroon" title="c14">b'</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:24_1_1_2_1_3"/>
<font color="Maroon" title="c14">b'</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c2">T</font>
 ;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c14">M</font> = <font color="Maroon" title="c7">M1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_1_2_1_4"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E2:24_1_1_2_1_4"/><b>then </b>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_2_1_4"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E15:24_1_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E16:24_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">M1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">M</font> = <font color="Maroon" title="c7">M1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_1_2_1_5"/><i><font color="Green" title="E28">A11</font></i>: 
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_2_1_5_1_1"/>
( <font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M1</font> or <font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_1_2_1_5"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_5_1_1_1"><b>case </b></a><a NAME="E1:24_1_1_2_1_5_1_1_1"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1_2_1_5_1_1_1"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E4:24_1_1_2_1"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_5_1_1_2"><b>case </b></a><a NAME="E1:24_1_1_2_1_5_1_1_2"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:24_1_1_2_1_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E11:24_1_1_2_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_2_1_5_1_1_2"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:24_1_1_2_1_5"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E19:24_1_1_2_1"/><b>then </b><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon" title="c15">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E20:24_1_1_2_1"/>
not <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M1</font>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E11:24_1_1_2_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E21:24_1_1_2_1"/><b>then </b><i><font color="Green" title="E29">A13</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="groeb_3.html#E4:24_1_1_2_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">b1</font>, <font color="Maroon" title="c17">b2</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="E1:24_1_1_2_1_6"/><i><font color="Green" title="E30">A14</font></i>: 
( <font color="Maroon" title="c16">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c17">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c16">b1</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c17">b2</font>,<font color="Maroon" title="c2">T</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_2_1_6_1_1"/>
( <font color="Maroon" title="c16">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M1</font> or <font color="Maroon" title="c16">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_1_2_1_6"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_6_1_1_1"><b>case </b></a><a NAME="E1:24_1_1_2_1_6_1_1_1"/>
<font color="Maroon" title="c16">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M1</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_2_1_6_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M1</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E4:24_1_1_2_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="groeb_3.html#E1:24_1_1_2_1_6"><i><font color="Green" title="E30">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_2_1_6_1_1_1"/>
<font color="Maroon" title="c17">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_6_1_1_2"><b>case </b></a><a NAME="E1:24_1_1_2_1_6_1_1_2"/>
<font color="Maroon" title="c16">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:24_1_1_2_1_6_1_1_2"/><b>then </b><i><font color="Green" title="E31">A15</font></i>: 
( <font color="Maroon" title="c16">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c16">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font> )
 <b>by </b><i><a class="txt" href="groeb_3.html#E11:24_1_1_2_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1_6_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_6_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="E1:24_1_1_2_1_6_1_1_2_1_1"/>
( <font color="Maroon" title="c17">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b1</font> or <font color="Maroon" title="c17">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_6_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:24_1_1_2_1_6_1_1_2_1_1_1"/>
<font color="Maroon" title="c17">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1_2_1_6_1_1_2_1_1_1"/>
<font color="Maroon" title="c17">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_1_2_1_6"><i><font color="Green" title="E30">A14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_6_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:24_1_1_2_1_6_1_1_2_1_1_2"/>
<font color="Maroon" title="c17">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">b1</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_2_1_6_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E32">A16</font></i>: 
<font color="Maroon" title="c16">b1</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c17">b2</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_1_2_1_6"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1_6_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_1_2_1_6_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c17">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font>
 ;<br/><a NAME="E2:24_1_1_2_1_6_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">b2</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c16">b1</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E13:24_1_1_2_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="groeb_3.html#E2:24_1_1_2_1_6_1_1_2"><i><font color="Green" title="E31">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_2_1_6_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:24_1_1_2_1_6_1_1_2_1_1_2"><i><font color="Green" title="E32">A16</font></i></a>, <a class="ref" href="termord.html#T5" title="TERMORD:th.5">TERMORD:5</a></i>;<br/></div><b>end;</b></div><a NAME="E4:24_1_1_2_1_6_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M1</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_1_2_1_6"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:24_1_1_2_1_6_1_1_2_1_1_2"/>
<font color="Maroon" title="c17">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_1_2_1_6_1_1_2"/>
<font color="Maroon" title="c17">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:24_1_1_2_1_6"/>
<font color="Maroon" title="c17">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E23:24_1_1_2_1"/>
 ex <font color="Olive" title="b1">M</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">b'</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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b2">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b3">b'</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b3">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> ) )
 <b>by </b><i><a class="txt" href="groeb_3.html#E19:24_1_1_2_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="groeb_3.html#E21:24_1_1_2_1"><i><font color="Green" title="E29">A13</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:24_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:24_1_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="groeb_3.html#E1:24_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="groeb_3.html#E2:24_1_1"><i><font color="Green" title="E21">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:24_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> &amp; ( for <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">b'</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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive" title="b3">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b2">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b3">b'</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b3">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>


</div>
