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<b>let </b><font color="Maroon">IT1</font> be   <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>, <font color="Maroon">IT2</font> be   <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39_1_2"/><i><font color="Green">E163</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT1</font> iff ( <font color="Olive">p</font> <a href="pboole.html#R6">=</a> <font color="Olive">q</font> or  ex <font color="Olive">i</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Olive">p</font> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">q</font> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">k</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> holds <br/><font color="Olive">p</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> ) ) ) )
 <b>and </b><br/><a NAME="E2:39_1_2"/><i><font color="Green">E164</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT2</font> iff ( <font color="Olive">p</font> <a href="pboole.html#R6">=</a> <font color="Olive">q</font> or  ex <font color="Olive">i</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Olive">p</font> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">q</font> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">k</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> holds <br/><font color="Olive">p</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:39_1_2_1_1"/><i><font color="Green">E165</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT1</font>
 ;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_2_1_1"/><i><font color="Green">E166</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font>, <font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="bagorder.html#T1" target="_self">Th1</a>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_2_1_1_1"/>
( <font color="Maroon">p</font> <a href="pboole.html#R6">=</a> <font color="Maroon">q</font> or <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="pboole.html#R6">=</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_2_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT2</font>
 <b>by </b><i>, <a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_2_1_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><a NAME="E2:39_1_2_1_1_1_2"/><b>then </b>
 ex <font color="Olive">i</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">k</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> ) )
 <b>by </b><i>, , <a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:39_1_2_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT2</font>
 <b>by </b><i>, <a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:39_1_2_1"/><i><font color="Green">E167</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT2</font>
 ;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:39_1_2_1"/><i><font color="Green">E168</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font>, <font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="bagorder.html#T1" target="_self">Th1</a>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_2_1_2"/>
( <font color="Maroon">p</font> <a href="pboole.html#R6">=</a> <font color="Maroon">q</font> or <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_2_1"><b>suppose </b></a><a NAME="E1:39_1_2_1_2_1"/>
<font color="Maroon">p</font> <a href="pboole.html#R6">=</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_2_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT1</font>
 <b>by </b><i>, <a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_2_2"><b>suppose </b></a><a NAME="E1:39_1_2_1_2_2"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><a NAME="E2:39_1_2_1_2_2"/><b>then </b>
 ex <font color="Olive">i</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">k</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> ) )
 <b>by </b><i>, , <a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:39_1_2_1_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT1</font>
 <b>by </b><i>, <a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39_1_2"/>
<font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/>


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