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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/>



<b>assume </b><a NAME="E1:105"/><i><font color="Green" title="E52">A1</font></i>: 
not <font color="Maroon" title="c2">p</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c3">q</font>
 ;<br/>

<a NAME="E2:105"/><b>then </b><i><font color="Green" title="E53">A2</font></i>: 
( not <font color="Maroon" title="c2">p</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font> )
 
<b>by </b><i><a class="ref" href="polynom1.html#D12" target="_self" title="POLYNOM1:def.12">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:105"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E5:105"/><i><font color="Green" title="E55">A4</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:105"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">j</font> = <font color="Maroon" title="c4">i</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="polynom1.html#E4:105"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon" title="c2">p</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> $1;<br/>
<a NAME="E7:105"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E4:105"><i><font color="Green" title="E54">A3</font></i></a></i>;<br/>
<a NAME="E8:105"/><b>then </b><i><font color="Green" title="E56">A5</font></i>: 
 ex <font color="Olive" title="b1">i</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>by </b><i><a class="txt" href="polynom1.html#E5:105"><i><font color="Green" title="E55">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">m</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E10:105"/><i><font color="Green" title="E57">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font>]
 <b>and </b><br/><a NAME="E11:105"/><i><font color="Green" title="E58">A7</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c6">m</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">n</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="polynom1.html#E8:105"><i><font color="Green" title="E56">A5</font></i></a>);<br/></i>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1"/>
( <font color="Maroon" title="c2">p</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">m</font> or <font color="Maroon" title="c2">p</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">q</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">m</font> )
 </span><b>by </b><i><a class="txt" href="polynom1.html#E10:105"><i><font color="Green" title="E57">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1"><b>suppose </b></a><a NAME="E1:105_1_1"/><i><font color="Green" title="E59">A8</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">m</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">l</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:105_1_1_1"/>
<font color="Maroon" title="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font>
 ;<br/><a NAME="E2:105_1_1_1"/><b>then </b>
not <font color="Maroon" title="c6">m</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">l</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/><b>hence </b><a NAME="E3:105_1_1_1"/>
<font color="Maroon" title="c3">q</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">l</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E11:105"><i><font color="Green" title="E58">A7</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:105_1_1"/>
<font color="Maroon" title="c3">q</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:105"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="polynom1.html#E1:105_1_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="polynom1.html#D11" target="_self" title="POLYNOM1:def.11">Def11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2"><b>suppose </b></a><a NAME="E1:105_1_2"/><i><font color="Green" title="E59">A9</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">q</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">m</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">l</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:105_1_2_1"/>
<font color="Maroon" title="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font>
 ;<br/><a NAME="E2:105_1_2_1"/><b>then </b>
not <font color="Maroon" title="c6">m</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">l</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/><b>hence </b><a NAME="E3:105_1_2_1"/>
<font color="Maroon" title="c2">p</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">l</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E11:105"><i><font color="Green" title="E58">A7</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:105_1_2"/><b>then </b>
<font color="Maroon" title="c3">q</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:105_1_2"><i><font color="Green" title="E59">A9</font></i></a>, <a class="ref" href="polynom1.html#D11" target="_self" title="POLYNOM1:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E4:105_1_2"/>
<font color="Maroon" title="c3">q</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D12" target="_self" title="POLYNOM1:def.12">Def12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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