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<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>





<a NAME="E1:19"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">T</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1"/>
( <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>suppose </b></a><a NAME="E1:19_1_1"/>
<font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_1"/>
( <font color="Maroon">b1</font> <a href="termord.html#R1" target="_self">&lt;=</a> <font color="Maroon">b2</font>,<font color="Maroon">T</font> iff not <font color="Maroon">b2</font> <a href="termord.html#R2" target="_self">&lt;</a> <font color="Maroon">b1</font>,<font color="Maroon">T</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>suppose </b></a><a NAME="E1:19_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:19_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">b1</font> <a href="termord.html#R1" target="_self">&lt;=</a> <font color="Maroon">b2</font>,<font color="Maroon">T</font> implies  not <font color="Maroon">b2</font> <a href="termord.html#R2" target="_self">&lt;</a> <font color="Maroon">b1</font>,<font color="Maroon">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">b1</font> <a href="termord.html#R1" target="_self">&lt;=</a> <font color="Maroon">b2</font>,<font color="Maroon">T</font>
 ;<br/>

<b>assume </b><a NAME="E2:19_1_2_1"/>
<font color="Maroon">b2</font> <a href="termord.html#R2" target="_self">&lt;</a> <font color="Maroon">b1</font>,<font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:19_1_2_1"/><b>then </b>
( <font color="Maroon">b2</font> <a href="termord.html#R1" target="_self">&lt;=</a> <font color="Maroon">b1</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:19_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E3:19_1_2"/>
( not <font color="Maroon">b2</font> <a href="termord.html#R2" target="_self">&lt;</a> <font color="Maroon">b1</font>,<font color="Maroon">T</font> implies <font color="Maroon">b1</font> <a href="termord.html#R1" target="_self">&lt;=</a> <font color="Maroon">b2</font>,<font color="Maroon">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_2_2"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">b2</font> <a href="termord.html#R2" target="_self">&lt;</a> <font color="Maroon">b1</font>,<font color="Maroon">T</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_2_2_1_1"/>
(  not <font color="Maroon">b2</font> <a href="termord.html#R1" target="_self">&lt;=</a> <font color="Maroon">b1</font>,<font color="Maroon">T</font> or <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2</font> )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_2_1_1_1"><b>case </b></a><a NAME="E1:19_1_2_2_1_1_1"/>
not <font color="Maroon">b2</font> <a href="termord.html#R1" target="_self">&lt;=</a> <font color="Maroon">b1</font>,<font color="Maroon">T</font>
 ;<br/><div class="add"><a NAME="E2:19_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i/>;<br/><a NAME="E3:19_1_2_2_1_1_1"/>
( <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">T</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">T</font> )
 <b>by </b><i/>;<br/><a NAME="E4:19_1_2_2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i>, , , <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/><b>hence </b><a NAME="E5:19_1_2_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="termord.html#R1" target="_self">&lt;=</a> <font color="Maroon">b2</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_2_1_1_2"><b>case </b></a><a NAME="E1:19_1_2_2_1_1_2"/>
<font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_2_2_1_1_2"/>
<font color="Maroon">b1</font> <a href="termord.html#R1" target="_self">&lt;=</a> <font color="Maroon">b2</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_1_2_2"/>
<font color="Maroon">b1</font> <a href="termord.html#R1" target="_self">&lt;=</a> <font color="Maroon">b2</font>,<font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:19_1_2"/>
( <font color="Maroon">b1</font> <a href="termord.html#R1" target="_self">&lt;=</a> <font color="Maroon">b2</font>,<font color="Maroon">T</font> iff not <font color="Maroon">b2</font> <a href="termord.html#R2" target="_self">&lt;</a> <font color="Maroon">b1</font>,<font color="Maroon">T</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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