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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:51"/><i><font color="Green">E30</font></i>: 
 <a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R7">is_strongly_connected_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="bagorder.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E2:51"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E3:51"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E4:51"/><i><font color="Green">E33</font></i>: 
 <a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E2:51"><i><font color="Green">E31</font></i></a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<a NAME="E5:51"/><i><font color="Green">E34</font></i>: 
 <a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E2:51"><i><font color="Green">E31</font></i></a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a></i>;<br/>
<a NAME="E6:51"/><i><font color="Green">E35</font></i>: 
 <a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:51"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bagorder.html#E2:51"><i><font color="Green">E31</font></i></a>, <a class="ref" href="orders_1.html#T92">ORDERS_1:92</a></i>;<br/>
<a NAME="E7:51"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K9">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="bagorder.html#D7" target="_self">Def7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:51_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:51_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:51_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:51_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="bagorder.html#E1:51_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bagorder.html#E4:51_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/><a NAME="E6:51_1"/><i><font color="Green">E40</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="bagorder.html#E4:51_1"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E7:51_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:51_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E4:51_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<b>thus </b><a NAME="E5:51_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E3:51_1_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="bagorder.html#E6:51_1"><i><font color="Green">E40</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> ;<br/><a NAME="E9:51_1"/><i><font color="Green">E41</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:51_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E4:51_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:51_1_2"/>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>
<b>thus </b><a NAME="E6:51_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:51"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bagorder.html#E3:51_1_2"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:51_1"/>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:51_1"/><i><font color="Green">E42</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E13:51_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E14:51_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="bagorder.html#E12:51_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="bagorder.html#E13:51_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E15:51_1"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/><a NAME="E16:51_1"/>
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> is <a href="wellord1.html#V2">well-ordering</a>
 <b>by </b><i><a class="ref" href="bagorder.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E17:51_1"/><b>then </b>
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/><a NAME="E18:51_1"/><b>then </b>
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R1">is_well_founded_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:51_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E21:51_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="bagorder.html#E2:51"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bagorder.html#E9:51_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="bagorder.html#E14:51_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="bagorder.html#E15:51_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/><a NAME="E22:51_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E21:51_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E24:51_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E25:51_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E20:51_1"><i><font color="Green">E46</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/><b>thus </b><a NAME="E26:51_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E24:51_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="bagorder.html#E25:51_1"><i><font color="Green">E50</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:51_1_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:51_1_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:51_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:51_1_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:51_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:51_1_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E4:51_1_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="bagorder.html#E1:51_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bagorder.html#E24:51_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="bagorder.html#E5:51_1_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_3_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>15</sub></font> or  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>15</sub></font> or  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>15</sub></font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_3_1_1"><b>suppose </b></a><a NAME="E1:51_1_3_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><a NAME="E2:51_1_3_1_1"/>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:51_1_3"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E3:51_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E24:51_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="bagorder.html#E25:51_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bagorder.html#E1:51_1_3_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_3_1_2"><b>suppose </b></a><a NAME="E1:51_1_3_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><a NAME="E2:51_1_3_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E7:51"><i><font color="Green">E36</font></i></a>, <a class="txt" href="bagorder.html#E6:51_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="bagorder.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E3:51_1_3_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E6:51_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E4:51_1_3_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="bagorder.html#E24:51_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="bagorder.html#E25:51_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bagorder.html#E5:51_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bagorder.html#E1:51_1_3_1_2"><i><font color="Green">E55</font></i></a></i>;<br/><b>hence </b><a NAME="E5:51_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E22:51_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bagorder.html#E3:51_1_3_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_3_1_3"><b>suppose </b></a><a NAME="E1:51_1_3_1_3"/>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:51_1_3_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E7:51"><i><font color="Green">E36</font></i></a>, <a class="txt" href="bagorder.html#E6:51_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="bagorder.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E28:51_1"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:51"/><b>then </b>
 <a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R1">is_well_founded_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/>
<a NAME="E10:51"/><b>then </b>
 <a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E2:51"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bagorder.html#E3:51"><i><font color="Green">E32</font></i></a>, <a class="txt" href="bagorder.html#E4:51"><i><font color="Green">E33</font></i></a>, <a class="txt" href="bagorder.html#E5:51"><i><font color="Green">E34</font></i></a>, <a class="txt" href="bagorder.html#E6:51"><i><font color="Green">E35</font></i></a>, <a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:51"/>
 <a href="bagorder.html#K7" target="_self">GrInvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> is <a href="wellord1.html#V2">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>


</div>
