<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">o</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="E1:44"/><i><font color="Green">E30</font></i>: 
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> <font color="Maroon">o</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> <font color="Maroon">o</font>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">o</font> <a href="relat_2.html#R7">is_strongly_connected_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">y</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">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="txt" href="bagorder.html#E1:44"><i><font color="Green">E30</font></i></a>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>assume </b><a NAME="E3:44_1"/><i><font color="Green">E38</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K5" target="_self">Graded</a> <font color="Maroon">o</font>
 ;<br/><a NAME="E4:44_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">x'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">y'</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">x'</font> or  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">x'</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1"/>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">y'</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">x'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K5" target="_self">Graded</a> <font color="Maroon">o</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="44_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">x'</font>
 ;<br/><div class="add"><a NAME="E2:44_1_1_2"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="txt" href="bagorder.html#E2:44"><i><font color="Green">E31</font></i></a>, <a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:44_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="txt" href="bagorder.html#E1:44"><i><font color="Green">E30</font></i></a>, <a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K5" target="_self">Graded</a> <font color="Maroon">o</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="E4:44"/>
 <a href="bagorder.html#K5" target="_self">Graded</a> <font color="Maroon">o</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="44_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="E1:44_2"/><i><font color="Green">E50</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_1"/>
( <font color="Maroon">a</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_1"><b>suppose </b></a><a NAME="E1:44_2_1_1"/>
<font color="Maroon">a</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_2_1_1"/>
<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="bagorder.html#K5" target="_self">Graded</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T12">ORDERS_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_2"><b>suppose </b></a><a NAME="E1:44_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:44_2_1_2"/><b>then </b>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:44_2_1_2"/><b>then </b>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:44"><i><font color="Green">E30</font></i></a></i>;<br/><b>hence </b><a NAME="E4:44_2_1_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="bagorder.html#K5" target="_self">Graded</a> <font color="Maroon">o</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="E6:44"/>
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#K5" target="_self">Graded</a> <font color="Maroon">o</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_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="E1:44_3"/><i><font color="Green">E62</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="bagorder.html#K5" target="_self">Graded</a> <font color="Maroon">o</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_3_1"/>
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">b</font> or (  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">b</font> &amp; <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">o</font> ) )
 </span><b>by </b><i>, <a class="txt" href="bagorder.html#E1:44"><i><font color="Green">E30</font></i></a>, <a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_3_1_1"><b>suppose </b></a><a NAME="E1:44_3_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:44_3_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:44_3_1_1"/>
 <a href="bagorder.html#K3" target="_self">TotDegree</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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:44_3_1_1"/><b>then </b>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <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="xxreal_0.html#R1">&lt;</a>  <a href="bagorder.html#K3" target="_self">TotDegree</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>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:44"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:44_3_1_1"/>
<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="bagorder.html#K5" target="_self">Graded</a> <font color="Maroon">o</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="44_3_1_2"><b>suppose </b></a><a NAME="E1:44_3_1_2"/><i><font color="Green">E68</font></i>: 
(  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">b</font> &amp; <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">o</font> )
 ;<br/><div class="add"><a NAME="E2:44_3_1_2"/><b>then </b>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:44_3_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <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="hidden.html#R1">=</a>  <a href="bagorder.html#K3" target="_self">TotDegree</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>
 <b>by </b><i/>;<br/><a NAME="E4:44_3_1_2"/>
<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> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="txt" href="bagorder.html#E2:44"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E5:44_3_1_2"/>
<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="bagorder.html#K5" target="_self">Graded</a> <font color="Maroon">o</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="E8:44"/>
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#K5" target="_self">Graded</a> <font color="Maroon">o</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#K5" target="_self">Graded</a> <font color="Maroon">o</font>
 ;<br/>


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