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<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="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">L</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</a> <br/> for <font color="Olive" title="b3">m</font> being  <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font> holds <br/> (  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b3">m</font>,<font color="Olive" title="b1">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Olive" title="b3">m</font> &amp;  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Olive" title="b3">m</font>,<font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Olive" title="b3">m</font> &amp;  <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Olive" title="b3">m</font>,<font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">m</font> )</span><br/><b>let </b><font color="Maroon" title="c2">O</font> be  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <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">L</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</a> <br/> for <font color="Olive" title="b2">m</font> being  <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font> holds <br/> (  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b2">m</font>,<font color="Maroon" title="c2">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Olive" title="b2">m</font> &amp;  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Olive" title="b2">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Olive" title="b2">m</font> &amp;  <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Olive" title="b2">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> )</span><br/><b>let </b><font color="Maroon" title="c3">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</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">m</font> being  <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> holds <br/> (  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Olive" title="b1">m</font> &amp;  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Olive" title="b1">m</font> &amp;  <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> )</span><br/><b>let </b><font color="Maroon" title="c4">m</font> be   <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font> &amp;  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c4">m</font> &amp;  <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )</span><br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_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"/>
(  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="ref" href="polynom7.html#T6" title="POLYNOM7:th.6">POLYNOM7:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font> &amp;  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c4">m</font> &amp;  <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )</span><br/><div class="add"><b>hence </b><i><font color="Green" title="E28">A2</font></i>:  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font> = 
 <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="polynom7.html#D6" title="POLYNOM7:def.6">POLYNOM7:def 6</a></i>
<br/>.= 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font>
<b>by </b><i><a class="txt" href="termord.html#E1:39_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="termord.html#D6" target="_self" title="TERMORD:def.6">Def6</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c4">m</font> &amp;  <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )</span><br/><b>hence </b><a NAME="E3:39_1_1"/>
 <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="polynom7.html#D7" title="POLYNOM7:def.7">POLYNOM7:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font></span><br/><b>hence </b><a NAME="E4:39_1_1"/>
 <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="termord.html#E2:39_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="polynom7.html#T11" title="POLYNOM7:th.11">POLYNOM7:11</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="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/><i><font color="Green" title="E27">A3</font></i>: 
 ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font> &amp;  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c4">m</font> &amp;  <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">b</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:39_1_2"/><i><font color="Green" title="E28">A4</font></i>: 
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E4:39_1_2"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="termord.html#E3:39_1_2"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:39_1_2"/><b>then </b><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E6:39_1_2"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="termord.html#E1:39_1_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="termord.html#D6" target="_self" title="TERMORD:def.6">Def6</a></i>;<br/><b>hence </b><i><font color="Green" title="E30">A6</font></i>:  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> = 
<font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="txt" href="termord.html#E3:39_1_2"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font>
<b>by </b><i><a class="txt" href="termord.html#E5:39_1_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="polynom7.html#D6" title="POLYNOM7:def.6">POLYNOM7:def 6</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c4">m</font> &amp;  <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )</span><br/><b>hence </b><a NAME="E8:39_1_2"/>
 <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="polynom7.html#D7" title="POLYNOM7:def.7">POLYNOM7:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font></span><br/><b>hence </b><a NAME="E9:39_1_2"/>
 <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="termord.html#E7:39_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="polynom7.html#T11" title="POLYNOM7:th.11">POLYNOM7:11</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>

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