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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">p</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">q</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>



<b>assume </b><a NAME="E1:106"/><i><font color="Green">E41</font></i>: 
not <font color="Maroon">p</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:106"/><b>then </b><i><font color="Green">E42</font></i>: 
( not <font color="Maroon">p</font> <a href="polynom1.html#R1" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; not <font color="Maroon">p</font> <a href="pboole.html#R6">=</a> <font color="Maroon">q</font> )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:106"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E5:106"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="polynom1.html#T5" target="_self">Th5</a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">p</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E7:106"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E8:106"/><b>then </b><i><font color="Green">E45</font></i>: 
 ex <font color="Olive">i</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E10:106"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>]
 <b>and </b><br/><a NAME="E11:106"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">m</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">n</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:106_1"/>
( <font color="Maroon">p</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">m</font> or <font color="Maroon">p</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">m</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1"><b>suppose </b></a><a NAME="E1:106_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:106_1_1_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:106_1_1_1"/><b>then </b>
not <font color="Maroon">m</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/><b>hence </b><a NAME="E3:106_1_1_1"/>
<font color="Maroon">q</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:106_1_1"/>
<font color="Maroon">q</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_2"><b>suppose </b></a><a NAME="E1:106_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">p</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:106_1_2_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:106_1_2_1"/><b>then </b>
not <font color="Maroon">m</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/><b>hence </b><a NAME="E3:106_1_2_1"/>
<font color="Maroon">p</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><a NAME="E3:106_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="polynom1.html#R1" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:106_1_2"/>
<font color="Maroon">q</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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