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<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_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/><b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> )
 ;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E3:35_1"/>
( <font color="Maroon">a'</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">b'</font> or <font color="Maroon">b'</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">a'</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T49">POLYNOM1:49</a></i>;<br/><b>hence </b><a NAME="E4:35_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>  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#D16">POLYNOM1:def 16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:35"/>
 <a href="bagorder.html#NK4" target="_self">LexOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R7">is_strongly_connected_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="bagorder.html#D7" target="_self">BAGORDER:def 7</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><a NAME="E1:35_2"/>
 <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T64">POLYNOM1:64</a></i>;<br/><b>hence </b><a NAME="E2:35_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D16">POLYNOM1:def 16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:35"/>
for <font color="Olive">a</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span>,<font color="Olive">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#NK4" target="_self">LexOrder</a> <font color="Maroon">n</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">c</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><b>assume </b><a NAME="E1:35_3"/><i><font color="Green">E31</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>  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:35_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D16">POLYNOM1:def 16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_3_1_1"/>
( <font color="Maroon">a</font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i>, <a class="ref" href="polynom1.html#D12">POLYNOM1:def 12</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_1"><b>suppose </b></a><a NAME="E1:35_3_1_1_1"/>
<font color="Maroon">a</font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:35_3_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E4:35_3_1_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">l</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Olive">l</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D11">POLYNOM1:def 11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><a NAME="E1:35_3_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E2:35_3_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:35_3_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:35_3_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>let </b><font color="Maroon">l</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E5:35_3_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E6:35_3_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">l</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">l</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>;<br/><b>hence </b><a NAME="E7:35_3_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><a NAME="E6:35_3_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D11">POLYNOM1:def 11</a></i>;<br/><b>hence </b><a NAME="E7:35_3_1_1_1"/>
<font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D12">POLYNOM1:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_2"><b>suppose </b></a><a NAME="E1:35_3_1_1_2"/>
<font color="Maroon">a</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_3_1_1_2"/>
<font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:35_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D16">POLYNOM1:def 16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:35"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#NK4" target="_self">LexOrder</a> <font color="Maroon">n</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="polynom1.html#K9">+</a> <font color="Olive">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b</font> <a href="polynom1.html#K9">+</a> <font color="Olive">c</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#NK4" target="_self">LexOrder</a> <font color="Maroon">n</font>
 ;<br/>


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