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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:4"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:4_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="polyred.html#E1:4"><i><font color="Green">E3</font></i></a>, <a class="ref" href="polynom1.html#D12">POLYNOM1:def 12</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:4_1_2"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D11">POLYNOM1:def 11</a></i>;<br/><a NAME="E5:4_1_2"/><i><font color="Green">E5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>;<br/><a NAME="E6:4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>;<br/><a NAME="E7:4_1_2"/><b>then </b><i><font color="Green">E6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="polyred.html#E4:4_1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="polyred.html#E5:4_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xreal_1.html#T10">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">c<sub>6</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:4_1_2_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E4:4_1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="polyred.html#E1:4_1_2_1"><i><font color="Green">E7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div><a NAME="E9:4_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="polyred.html#E7:4_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="polynom1.html#D11">POLYNOM1:def 11</a></i>;<br/><b>hence </b><a NAME="E10:4_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D12">POLYNOM1:def 12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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