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

<b>set </b><font color="Maroon" title="c5">O</font> = <font color="Maroon" title="c2">T</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_1"/>
(  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1_1"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> st <br/>( (  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font> ) or ( <font color="Olive">b<sub>1</sub></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; ( for <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="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> holds <br/><font color="Olive" title="b2">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c2">T</font> ) ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_1_2"/><i><font color="Green" title="E21">A1</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <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> 
 ;<br/><div class="add"><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>for <font color="Olive" title="b1">B</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  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <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 <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> &amp; ( for <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="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">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> ) );<br/><a NAME="E2:31_1_1_1_2"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">B</font> be  <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>assume </b><a NAME="E1:31_1_1_1_2_1"/><i><font color="Green" title="E23">A3</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>
<a NAME="E3:31_1_1_1_2_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="termord.html#E1:31_1_1_1_2_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:31_1_1_1_2_1"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="card_1.html#T49" title="CARD_1:th.49">CARD_1:49</a></i>;<br/>
<a NAME="E6:31_1_1_1_2_1"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="termord.html#E5:31_1_1_1_2_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c8">b</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c9">b</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">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="E1:31_1_1_1_2_1_1"/>
<font color="Maroon" title="c11">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 ;<br/><a NAME="E2:31_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c11">b'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="termord.html#E5:31_1_1_1_2_1"><i><font color="Green" title="E24">A4</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="E3:31_1_1_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c10">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="ref" href="orders_1.html#T12" title="ORDERS_1:th.12">ORDERS_1:12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:31_1_1_1_2_1"/>
 ex <font color="Olive" title="b1">b'</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">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; ( for <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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">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="termord.html#E6:31_1_1_1_2_1"><i><font color="Green" title="E25">A5</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:31_1_1_1_2"/><i><font color="Green" title="E23">A6</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <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="31_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:31_1_1_1_2_2"/><i><font color="Green" title="E24">A7</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 ;<br/>

<b>thus </b><a NAME="E2:31_1_1_1_2_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="31_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_1_1_2_2_1"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ;<br/>

<b>thus </b><a NAME="E2:31_1_1_1_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="31_1_1_1_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">B</font> be  <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>assume </b><a NAME="E1:31_1_1_1_2_2_1_1"/><i><font color="Green" title="E26">A9</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c7">B</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> <b>by </b><i><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">x</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of <font color="Maroon" title="c8">B</font>;<br/>
<a NAME="E4:31_1_1_1_2_2_1_1"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c8">B</font> <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> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">x</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> ;<br/>
<b>set </b><font color="Maroon" title="c12">X</font> = <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E7:31_1_1_1_2_2_1_1"/><i><font color="Green" title="E27">A10</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> iff ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; not <font color="Maroon" title="c11">x</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="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 
<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><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:31_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c13">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="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E2:31_1_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</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="E3:31_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E9:31_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E28">A11</font></i>: <font color="Maroon" title="c8">B</font> = 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
;<br/>
<a NAME="E11:31_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E29">A12</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="termord.html#E1:31_1_1_1_2_2_1_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="termord.html#E7:31_1_1_1_2_2_1_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">X</font> = <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="termord.html#E1:31_1_1_1_2_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">b'</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="E14:31_1_1_1_2_2_1_1"/><i><font color="Green" title="E30">A13</font></i>: 
( <font color="Maroon" title="c14">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> &amp; ( for <font color="Olive" title="b1">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="c13">X</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">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="termord.html#E1:31_1_1_1_2_2_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="termord.html#E11:31_1_1_1_2_2_1_1"><i><font color="Green" title="E29">A12</font></i></a></i>;<br/>
<a NAME="E15:31_1_1_1_2_2_1_1"/><i><font color="Green" title="E31">A14</font></i>: 
<font color="Maroon" title="c2">T</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_1_2_2_1_1_3_1"/>
( <font color="Maroon" title="c11">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b'</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">b'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_1"><b>case </b></a><a NAME="E1:31_1_1_1_2_2_1_1_3_1_1"/><i><font color="Green" title="E32">A15</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b'</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c15">b'</font> = <font color="Maroon" title="c14">b'</font>;<br/><a NAME="E2:31_1_1_1_2_2_1_1_3_1_1"/><i><font color="Green" title="E33">A16</font></i>: 
<font color="Maroon" title="c13">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">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="E1:31_1_1_1_2_2_1_1_3_1_1_1"/><i><font color="Green" title="E34">A17</font></i>: 
<font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_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="E1:31_1_1_1_2_2_1_1_3_1_1_1_1_1"/>
( <font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> or  not <font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:31_1_1_1_2_2_1_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_1_1_2_2_1_1_3_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c15">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="termord.html#E14:31_1_1_1_2_2_1_1"><i><font color="Green" title="E30">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:31_1_1_1_2_2_1_1_3_1_1_1_1_1_2"/>
not <font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font>
 ;<br/><div class="add"><a NAME="E2:31_1_1_1_2_2_1_1_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">b</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="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="termord.html#E10:31_1_1_1_2_2_1_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="termord.html#E1:31_1_1_1_2_2_1_1_3_1_1_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:31_1_1_1_2_2_1_1_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">x</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="E4:31_1_1_1_2_2_1_1_3_1_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c15">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="termord.html#E1:31_1_1_1_2_2_1_1_3_1_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="orders_1.html#T12" title="ORDERS_1:th.12">ORDERS_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:31_1_1_1_2_2_1_1_3_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c15">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>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:31_1_1_1_2_2_1_1_3_1_1"/>
 ex <font color="Olive" title="b1">b'</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">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; ( for <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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">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="termord.html#E14:31_1_1_1_2_2_1_1"><i><font color="Green" title="E30">A13</font></i></a>, <a class="txt" href="termord.html#E2:31_1_1_1_2_2_1_1_3_1_1"><i><font color="Green" title="E33">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2"><b>case </b></a><a NAME="E1:31_1_1_1_2_2_1_1_3_1_2"/><i><font color="Green" title="E32">A18</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">b'</font>
 ;<br/><div class="add"><a NAME="E2:31_1_1_1_2_2_1_1_3_1_2"/><i><font color="Green" title="E33">A19</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</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="ref" href="orders_1.html#T12" title="ORDERS_1:th.12">ORDERS_1:12</a></i>;<br/><a NAME="E3:31_1_1_1_2_2_1_1_3_1_2"/>
<font color="Maroon" title="c14">b'</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <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="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E4:31_1_1_1_2_2_1_1_3_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">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="ref" href="orders_1.html#T12" title="ORDERS_1:th.12">ORDERS_1:12</a></i>;<br/><a NAME="E5:31_1_1_1_2_2_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E34">A20</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c14">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="txt" href="termord.html#E2:31_1_1_1_2_2_1_1_3_1_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_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="E1:31_1_1_1_2_2_1_1_3_1_2_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</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> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c11">x</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> )
 </span><b>by </b><i><a class="txt" href="termord.html#E15:31_1_1_1_2_2_1_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="txt" href="termord.html#E1:31_1_1_1_2_2_1_1_3_1_2"><i><font color="Green" title="E32">A18</font></i></a>, <a class="txt" href="termord.html#E5:31_1_1_1_2_2_1_1_3_1_2"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_1"><b>case </b></a><a NAME="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_1"/><i><font color="Green" title="E35">A21</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</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>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:31_1_1_1_2_2_1_1_3_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">b'</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">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; ( for <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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">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> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c14">b'</font>
;<br/>

<a NAME="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1"/>
for <font color="Olive" title="b1">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="c8">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">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="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1"/><i><font color="Green" title="E36">A22</font></i>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_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="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c15">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font> or <font color="Maroon" title="c15">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 ;<br/><div class="add"><a NAME="E2:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1_1_1_1"/><b>then </b>
not <font color="Maroon" title="c15">b</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="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font>
 <b>by </b><i><a class="txt" href="termord.html#E1:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1"><i><font color="Green" title="E36">A22</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="E4:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">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="termord.html#E14:31_1_1_1_2_2_1_1"><i><font color="Green" title="E30">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c15">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">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="termord.html#E1:31_1_1_1_2_2_1_1_3_1_2_1_1_1"><i><font color="Green" title="E35">A21</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="E3:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">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>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:31_1_1_1_2_2_1_1_3_1_2_1_1_1_1"/>
( <font color="Maroon" title="c14">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; ( for <font color="Olive" title="b1">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="c8">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">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="termord.html#E14:31_1_1_1_2_2_1_1"><i><font color="Green" title="E30">A13</font></i></a>, <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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_2"><b>case </b></a><a NAME="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_2"/><i><font color="Green" title="E35">A23</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c11">x</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>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:31_1_1_1_2_2_1_1_3_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">b'</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">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; ( for <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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">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> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_2_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1"/>
for <font color="Olive" title="b1">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="c8">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">b</font>,<font color="Maroon" title="c11">x</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">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="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1"/><i><font color="Green" title="E36">A24</font></i>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_2_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="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c15">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font> or <font color="Maroon" title="c15">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c15">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 ;<br/><div class="add"><a NAME="E2:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1_1"/><b>then </b>
not <font color="Maroon" title="c15">b</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="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font>
 <b>by </b><i><a class="txt" href="termord.html#E1:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1"><i><font color="Green" title="E36">A24</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="E4:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E37">A25</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">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="termord.html#E14:31_1_1_1_2_2_1_1"><i><font color="Green" title="E30">A13</font></i></a></i>;<br/><a NAME="E5:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c15">b</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c14">b'</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <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="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>hence </b><a NAME="E6:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c11">x</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="termord.html#E1:31_1_1_1_2_2_1_1_3_1_2_1_1_2"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="termord.html#E4:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1_1"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="orders_1.html#T14" title="ORDERS_1:th.14">ORDERS_1:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c15">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c11">x</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="ref" href="orders_1.html#T12" title="ORDERS_1:th.12">ORDERS_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c11">x</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>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:31_1_1_1_2_2_1_1_3_1_2_1_1_2_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; ( for <font color="Olive" title="b1">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="c8">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">b</font>,<font color="Maroon" title="c11">x</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> ) )
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:31_1_1_1_2_2_1_1_3_1_2"/>
 ex <font color="Olive" title="b1">b'</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">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; ( for <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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">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> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:31_1_1_1_2_2_1_1"/>
 ex <font color="Olive" title="b1">b'</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">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; ( for <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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">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> ) )
 ;<br/>


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

<b>thus </b><a NAME="E3:31_1_1_1_2_2_1"/>
verum
 ;<br/>


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


</div><b>end;</b></div><a NAME="E4:31_1_1_1_2"/><i><font color="Green" title="E24">A26</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/><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#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="termord.html#E2:31_1_1_1_2"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="termord.html#E3:31_1_1_1_2"><i><font color="Green" title="E23">A6</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c6">sp</font> =  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></i>;<br/><a NAME="E6:31_1_1_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">sp</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:31_1_1_1_2"/><i><font color="Green" title="E25">A27</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">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>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="ref" href="card_1.html#T86" title="CARD_1:th.86">CARD_1:86</a></i>;<br/><a NAME="E9:31_1_1_1_2"/><i><font color="Green" title="E26">A28</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">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="c7">m</font>
 <b>by </b><i><a class="txt" href="termord.html#E8:31_1_1_1_2"><i><font color="Green" title="E25">A27</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1_1_1_2_3"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/><a NAME="E2:31_1_1_1_2_3"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">sp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>by </b><i><a class="txt" href="termord.html#E8:31_1_1_1_2"><i><font color="Green" title="E25">A27</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><b>hence </b><a NAME="E3:31_1_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="termord.html#E1:31_1_1_1_2"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/></div><b>end;</b></div><a NAME="E11:31_1_1_1_2"/><b>then </b><i><font color="Green" title="E27">A29</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">sp</font> =  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> 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> <b>by </b><i><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></i>;<br/><a NAME="E13:31_1_1_1_2"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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/><b>then consider </b><font color="Maroon" title="c9">b'</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="E15:31_1_1_1_2"/><i><font color="Green" title="E28">A30</font></i>: 
( <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">sp</font> &amp; ( for <font color="Olive" title="b1">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="c8">sp</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">b</font>,<font color="Maroon" title="c9">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="termord.html#E4:31_1_1_1_2"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="termord.html#E9:31_1_1_1_2"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="termord.html#E11:31_1_1_1_2"><i><font color="Green" title="E27">A29</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c9">b'</font>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">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="E1:31_1_1_1_2_4"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">sp</font>
 ;<br/><a NAME="E2:31_1_1_1_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c9">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="termord.html#E15:31_1_1_1_2"><i><font color="Green" title="E28">A30</font></i></a></i>;<br/><b>hence </b><a NAME="E3:31_1_1_1_2_4"/>
<font color="Maroon" title="c10">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#D2" target="_self" title="TERMORD:def.2">Def2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:31_1_1_1_2"/>
( <font color="Maroon" title="c9">b'</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> &amp; not <!--HUMANRECFAILED-->( not <!--HUMANRECFAILED-->(  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c9">b'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font> ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c9">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; ( for <font color="Olive" title="b1">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="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> holds <br/><font color="Olive" title="b1">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c2">T</font> ) ) ) )
 <b>by </b><i><a class="txt" href="termord.html#E15:31_1_1_1_2"><i><font color="Green" title="E28">A30</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
