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<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">T</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="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> <br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font> holds <br/> ( <font color="Olive" title="b3">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Olive" title="b4">q</font>,<font color="Olive" title="b1">T</font> iff not <font color="Olive" title="b4">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Olive" title="b3">p</font>,<font color="Olive" title="b1">T</font> )</span><br/><b>let </b><font color="Maroon" title="c2">T</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="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> <br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font> holds <br/> ( <font color="Olive" title="b2">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Olive" title="b3">q</font>,<font color="Maroon" title="c2">T</font> iff not <font color="Olive" title="b3">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Olive" title="b2">p</font>,<font color="Maroon" title="c2">T</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="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c2">T</font> iff not <font color="Olive" title="b2">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> )</span><br/><b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</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"> ( <font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font> iff not <font color="Maroon" title="c5">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font> )</span><br/>







<a NAME="E1:49"/><i><font color="Green" title="E40">A1</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font> implies  not <font color="Maroon" title="c5">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c5">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span><br/>

<b>assume </b><a NAME="E2:49_1"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:49_1"/><b>then </b>
<font color="Maroon" title="c5">q</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="polyred.html#D3" target="_self" title="POLYRED:def.3">Def3</a></i>;<br/>
<a NAME="E4:49_1"/><b>then </b>
 <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="#E1:49_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="polyred.html#T26" target="_self" title="POLYRED:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E5:49_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:49_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="polyred.html#D3" target="_self" title="POLYRED:def.3">Def3</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>
<a NAME="E2:49"/>
( not <font color="Maroon" title="c5">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_2"/><i><font color="Green" title="E41">A4</font></i>: 
not <font color="Maroon" title="c5">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1_1"/>
(  not  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">p</font> or  not <font color="Maroon" title="c5">q</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font> )
 </span><b>by </b><i><a class="txt" href="#E1:49_2"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="polyred.html#D3" target="_self" title="POLYRED:def.3">Def3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1"><b>case </b></a><a NAME="E1:49_2_1_1_1"/>
not  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c4">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span><br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#T26" target="_self" title="POLYRED:th.26">Th26</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="49_2_1_1_2"><b>case </b></a><a NAME="E1:49_2_1_1_2"/>
not <font color="Maroon" title="c5">q</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span><br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_1_2"/>
<font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#T28" target="_self" title="POLYRED:th.28">Th28</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="E3:49_2"/>
<font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font>
 ; <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>
<b>hence </b><a NAME="E3:49"/>
( <font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font> iff not <font color="Maroon" title="c5">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E40">A1</font></i></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>
