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

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o</font> being  <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>  st ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">o</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b4">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b4">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">o</font> ) &amp; <font color="Olive" title="b1">o</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Olive" title="b1">o</font> is <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a></span><br/><b>let </b><font color="Maroon" title="c2">o</font> be   <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b3">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font> ) &amp; <font color="Maroon" title="c2">o</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font> implies  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> is <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:43"/><i><font color="Green" title="E28">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b3">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font>
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c2">o</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> is <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font> &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> )</span><br/><b>assume </b><a NAME="E1:43_1"/><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c4">y</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="bagorder.html#E1:43_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>assume </b><a NAME="E3:43_1"/><i><font color="Green" title="E31">A4</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font></span><br/><a NAME="E4:43_1"/><b>then </b><i><font color="Green" title="E32">A5</font></i>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c5">x'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c6">y'</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="bagorder.html#D9" target="_self" title="BAGORDER:def.9">Def9</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_1"/>
(  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c6">y'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c5">x'</font> or  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c6">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c5">x'</font> )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E4:43_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1"/>
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c6">y'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c5">x'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font></span><br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="bagorder.html#D9" target="_self" title="BAGORDER:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_2"/><i><font color="Green" title="E33">A6</font></i>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c6">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c5">x'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font></span><br/><div class="add"><a NAME="E2:43_1_1_2"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="bagorder.html#E3:43_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="bagorder.html#D9" target="_self" title="BAGORDER:def.9">Def9</a></i>;<br/><a NAME="E3:43_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:43"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">RELAT_2:def 7</a></i>;<br/><b>hence </b><a NAME="E4:43_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="bagorder.html#E1:43_1_1_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="bagorder.html#D9" target="_self" title="BAGORDER:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43"/>
 <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">RELAT_2:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">BAGORDER:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b3">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font></span><br/><a NAME="E1:43_2"/><i><font color="Green" title="E30">A7</font></i>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="bagorder.html#T16" target="_self" title="BAGORDER:th.16">Th16</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_2_1"/>
( <font color="Maroon" title="c3">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1"><b>suppose </b></a><a NAME="E1:43_2_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font></span><br/><div class="add"><b>hence </b><a NAME="E2:43_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T12" title="ORDERS_1:th.12">ORDERS_1:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_2"><b>suppose </b></a><a NAME="E1:43_2_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font></span><br/><div class="add"><a NAME="E2:43_2_1_2"/><b>then </b>
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="bagorder.html#T16" target="_self" title="BAGORDER:th.16">Th16</a></i>;<br/><a NAME="E3:43_2_1_2"/><b>then </b>
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43_2"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E4:43_2_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="bagorder.html#D9" target="_self" title="BAGORDER:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:43"/>
for <font color="Olive" title="b1">a</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b3">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> )</span><br/><b>assume </b><a NAME="E1:43_3"/><i><font color="Green" title="E30">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_3_1"/>
(  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c4">b</font> or (  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c4">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font> ) )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="bagorder.html#E1:43_3"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="bagorder.html#D9" target="_self" title="BAGORDER:def.9">Def9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_3_1_1"><b>suppose </b></a><a NAME="E1:43_3_1_1"/><i><font color="Green" title="E31">A9</font></i>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font></span><br/><div class="add"><a NAME="E2:43_3_1_1"/><i><font color="Green" title="E32">A10</font></i>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c5">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#T14" target="_self" title="BAGORDER:th.14">Th14</a></i>;<br/><a NAME="E3:43_3_1_1"/>
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c5">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#T14" target="_self" title="BAGORDER:th.14">Th14</a></i>;<br/><a NAME="E4:43_3_1_1"/><b>then </b>
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43_3_1_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="bagorder.html#E2:43_3_1_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:43_3_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="bagorder.html#D9" target="_self" title="BAGORDER:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_3_1_2"><b>suppose </b></a><a NAME="E1:43_3_1_2"/><i><font color="Green" title="E31">A11</font></i>: 
(  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c4">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font></span><br/><div class="add"><a NAME="E2:43_3_1_2"/><b>then </b>
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c5">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#T14" target="_self" title="BAGORDER:th.14">Th14</a></i>;<br/><a NAME="E3:43_3_1_2"/><b>then </b><i><font color="Green" title="E32">A12</font></i>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#T14" target="_self" title="BAGORDER:th.14">Th14</a></i>;<br/><a NAME="E4:43_3_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="bagorder.html#E1:43_3_1_2"><i><font color="Green" title="E31">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E5:43_3_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="bagorder.html#E3:43_3_1_2"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="bagorder.html#D9" target="_self" title="BAGORDER:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:43"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b3">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K5" title="BAGORDER:func.5">Graded</a> <font color="Maroon" title="c2">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
