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



<b>assume </b><b>that </b><br/><a NAME="E1:111"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">d</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:111"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">d</font> <a href="polynom1.html#R1" target="_self">&lt;</a> <font color="Maroon">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="polynom1.html#D12" target="_self">POLYNOM1:def 12</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:111_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E3:111_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">d</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">p'</font>
 <b>by </b><i>, </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">d</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>assume </b><a NAME="E4:111_1"/>
<font color="Maroon">d</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E5:111_1"/><b>then </b>
<font color="Maroon">d</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">p'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">p'</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:111_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 ex <font color="Olive">p</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>]
 ;<br/><b>consider </b><font color="Maroon">k</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:111_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E9:111_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">m</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">k</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">m</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>(<a class="ref" href="polynom1.html#T7" target="_self">Th7</a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_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="E1:111_1_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:111_1_1"/><b>then </b>
not <font color="Maroon">k</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/><a NAME="E3:111_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">b</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/><a NAME="E4:111_1_1"/>
<font color="Maroon">d</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">l</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:111_1_1"/>
<font color="Maroon">d</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:111_1"/>
contradiction
 <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="E4:111"/>
<font color="Maroon">d</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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