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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>2</sub></font> or (  <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="bagorder.html#K3" target="_self">TotDegree</a> <font color="Olive">b<sub>2</sub></font> &amp; <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> <font color="Maroon">c<sub>2</sub></font> ) ) );<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="relset_1.html#NM3">Relation</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:43_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <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> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) )
 <b>from </b><i><a class="ref" href="relset_1.html#S1">RELSET_1:sch 1</a>();<br/></i>
<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_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><a NAME="E1:43_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="bagorder.html#E1:43_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><a NAME="E1:43_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>thus </b><a NAME="E2:43_1_1_1_1_1"/>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E3:43_1_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="orders_1.html#T12">ORDERS_1:12</a></i>;<br/><a NAME="E4:43_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E5:43_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T57">POLYNOM1:57</a></i>;<br/><b>hence </b><a NAME="E6:43_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T57">POLYNOM1:57</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:43_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:43_1_1_1"/>
<font color="Maroon">c<sub>3</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#D1">RELAT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1_1_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:43_1_1_1_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E4:43_1_1_1_2"/><i><font color="Green">E31</font></i>: 
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_2"><i><font color="Green">E29</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:43_1_1_1_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E7:43_1_1_1_2"/><i><font color="Green">E33</font></i>: 
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>9</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <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> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_2"><i><font color="Green">E29</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_1_1_2_1_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E4:43_1_1_1_2"><i><font color="Green">E31</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1_2_1_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_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="E1:43_1_1_1_2_1_1_1_1_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E3:43_1_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bagorder.html#E6:43_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="bagorder.html#E7:43_1_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1_2_1_1_1_1_1_1"/>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43_1_1_1_2_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_1_2_1_1_1_1_1_2"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43_1_1_1_2_1_1_1"><i><font color="Green">E34</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="E3:43_1_1_1_2_1_1_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_1_2_1_1_2"/><i><font color="Green">E34</font></i>: 
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_1_1_2_1_1_2_1_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E3:43_1_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bagorder.html#E6:43_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="bagorder.html#E7:43_1_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1_2_1_1_2_1_1_1"/>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_1_2_1_1_2_1_1_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43_1_1_1_2_1_1_2"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_1_2_1_1_2_1_1_2"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_1_2_1_1_2_1_1_2"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43_1_1_1_2_1_1_2"><i><font color="Green">E34</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="E3:43_1_1_1_2_1_1_2"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:43_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bagorder.html#E3:43_1_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="orders_1.html#T13">ORDERS_1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:43_1_1_1"/>
<font color="Maroon">c<sub>3</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="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1_1_1_3"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:43_1_1_1_3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_3"><i><font color="Green">E29</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:43_1_1_1_3"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>10</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_3"><i><font color="Green">E29</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_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="E1:43_1_1_1_3_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E3:43_1_1_1_3"><i><font color="Green">E30</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1_3_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_3_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="E1:43_1_1_1_3_1_1_1_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>10</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <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>10</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E3:43_1_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bagorder.html#E5:43_1_1_1_3"><i><font color="Green">E31</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1_3_1_1_1_1_1"/>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><a NAME="E2:43_1_1_1_3_1_1_1_1_1"/><b>then </b>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43_1_1_1_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:43_1_1_1_3_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bagorder.html#E3:43_1_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bagorder.html#E5:43_1_1_1_3"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_1_3_1_1_1_1_2"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <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>10</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_1_3_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bagorder.html#E3:43_1_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bagorder.html#E5:43_1_1_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_3_1_1"><i><font color="Green">E32</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="E3:43_1_1_1_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_1_3_1_2"/><i><font color="Green">E32</font></i>: 
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_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="E1:43_1_1_1_3_1_2_1_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>10</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <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>10</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E3:43_1_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bagorder.html#E5:43_1_1_1_3"><i><font color="Green">E31</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1_3_1_2_1_1_1"/>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_1_3_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bagorder.html#E3:43_1_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bagorder.html#E5:43_1_1_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_3_1_2"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_1_3_1_2_1_1_2"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <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>10</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><a NAME="E2:43_1_1_1_3_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43_1_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bagorder.html#E3:43_1_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bagorder.html#E5:43_1_1_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_3_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="orders_1.html#T14">ORDERS_1:14</a></i>;<br/><b>hence </b><a NAME="E3:43_1_1_1_3_1_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bagorder.html#E3:43_1_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bagorder.html#E5:43_1_1_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_1_3_1_2"><i><font color="Green">E32</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="E3:43_1_1_1_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:43_1_1_1"/>
<font color="Maroon">c<sub>3</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="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:43_1_1"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>3</sub></font> <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#T98">ORDERS_1:98</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="bagorder.html#E3:43_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:43_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:43_1_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E4:43_1_1_2"/><i><font color="Green">E30</font></i>: 
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43_1_1"><i><font color="Green">E27</font></i></a></i>;<br/><b>thus </b><a NAME="E5:43_1_1_2"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E3:43_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bagorder.html#E4:43_1_1_2"><i><font color="Green">E30</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E7:43_1_1"/><i><font color="Green">E29</font></i>: 
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/>

<a NAME="E8:43_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">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="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E9:43_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <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#E2:43_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E7:43_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>


</div>
