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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">b1</font>, <font color="Maroon" title="c3">b2</font>, <font color="Maroon" title="c4">b3</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="E1:4"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1"/>
( <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>suppose </b></a><a NAME="E1:4_1_1"/>
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1"/>
<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><b>suppose </b></a><a NAME="E1:4_1_2"/>
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b2</font>
 ;<br/><div class="add"><a NAME="E2:4_1_2"/><b>then </b>
<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c3">b2</font>
 <b>by </b><i><a class="txt" href="polyred.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="polynom1.html#D12" title="POLYNOM1:def.12">POLYNOM1:def 12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">k</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:4_1_2"/><i><font color="Green" title="E4">A2</font></i>: 
( <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font> &amp; ( 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="c5">k</font> holds <br/><font color="Maroon" title="c2">b1</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">b2</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/><a NAME="E5:4_1_2"/><i><font color="Green" title="E5">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b3</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>;<br/><a NAME="E6:4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b3</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>;<br/><a NAME="E7:4_1_2"/><b>then </b><i><font color="Green" title="E6">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="polyred.html#E4:4_1_2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="polyred.html#E5:4_1_2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">l</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:4_1_2_1"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c6">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font>
 ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b3</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">l</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b3</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">l</font></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E4:4_1_2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="polyred.html#E1:4_1_2_1"><i><font color="Green" title="E7">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c6">l</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div><a NAME="E9:4_1_2"/><b>then </b>
<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font>
 <b>by </b><i><a class="txt" href="polyred.html#E7:4_1_2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="polynom1.html#D11" title="POLYNOM1:def.11">POLYNOM1:def 11</a></i>;<br/><b>hence </b><a NAME="E10:4_1_2"/>
<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b3</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D12" title="POLYNOM1:def.12">POLYNOM1:def 12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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