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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">d</font>, <font color="Maroon" title="c3">b</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><b>that </b><br/><a NAME="E1:110"/><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E2:110"/><i><font color="Green" title="E58">A2</font></i>: 
not <font color="Maroon" title="c2">d</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c3">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="polynom1.html#D12" target="_self" title="POLYNOM1:def.12">POLYNOM1:def 12</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:110_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">p'</font> = <font color="Maroon" title="c4">p</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/><a NAME="E3:110_1"/><i><font color="Green" title="E59">A3</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">p'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">p'</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:110"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="polynom1.html#D13" target="_self" title="POLYNOM1:def.13">Def13</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">d</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> $1;<br/><b>assume </b><a NAME="E4:110_1"/>
<font color="Maroon" title="c2">d</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">p</font>
 ;<br/><a NAME="E5:110_1"/><b>then </b>
<font color="Maroon" title="c2">d</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">p'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">p'</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E3:110_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:110_1"/><b>then </b><i><font color="Green" title="E60">A4</font></i>: 
 ex <font color="Olive" title="b1">p</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">p</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c6">k</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E8:110_1"/><i><font color="Green" title="E61">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 <b>and </b><br/><a NAME="E9:110_1"/><i><font color="Green" title="E62">A6</font></i>: 
for <font color="Olive" title="b1">m</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">m</font>] holds <br/><font color="Maroon" title="c6">k</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">m</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#E6:110_1"><i><font color="Green" title="E60">A4</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_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="E1:110_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">k</font>
 ;<br/><a NAME="E2:110_1_1"/><b>then </b>
not <font color="Maroon" title="c6">k</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/><a NAME="E3:110_1_1"/><b>then </b><i><font color="Green" title="E63">A7</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">d</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#E9:110_1"><i><font color="Green" title="E62">A6</font></i></a></i>;<br/><a NAME="E4:110_1_1"/>
<font color="Maroon" title="c2">d</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</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#E1:110"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="polynom1.html#D13" target="_self" title="POLYNOM1:def.13">Def13</a></i>;<br/><b>hence </b><a NAME="E5:110_1_1"/>
<font color="Maroon" title="c2">d</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">b</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#E3:110_1_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:110_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E2:110"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="polynom1.html#E8:110_1"><i><font color="Green" title="E61">A5</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>
<b>hence </b><a NAME="E4:110"/>
<font color="Maroon" title="c2">d</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


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