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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">a</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><a NAME="E1:5"/>
<font color="Maroon" title="c2">a</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c3">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">o</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:5"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">o</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="c4">o</font>
 <b>and </b><br/><a NAME="E4:5"/><i><font color="Green" title="E4">A2</font></i>: 
for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">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="Olive" title="b1">l</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D11" title="POLYNOM1:def.11">POLYNOM1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">o</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1"/><i><font color="Green" title="E5">A3</font></i>: 
not <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 ;<br/><a NAME="E2:5_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:5_1"/><b>then </b><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="polynom6.html#E1:5_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/><a NAME="E4:5_1"/>
not <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:5_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E5:5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom6.html#E3:5"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="polynom6.html#E3:5_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:5"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>thus </b><a NAME="E7:5"/>
<font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">o</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="c4">o</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:5"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:5"/>
for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">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="Olive" title="b1">l</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:5"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>


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