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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM2" title="POLYNOM1:NM.2">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM2" title="POLYNOM1:NM.2">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">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/><span class="kw">let </span><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
(  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_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:49_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:49_1_1_1"/>
(  not  <a href="polynom1.html#K2" title="POLYNOM1:func.2">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#K2" title="POLYNOM1:func.2">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="polyred.html#D3" target="_self" title="POLYRED:def.3">Def3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:49_1_1_1_1"/>
 not  <a href="polynom1.html#K2" title="POLYNOM1:func.2">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c4">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">hence </span><a NAME="E2:49_1_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T26" target="_self" title="POLYRED:th.26">Th26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:49_1_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">hence </span><a NAME="E2:49_1_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T28" target="_self" title="POLYRED:th.28">Th28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:49_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:49"/>
( <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> 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_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:49_2"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E2:49_2"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:49_2"/><span class="kw">then </span>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D3" target="_self" title="POLYRED:def.3">Def3</a></span>;<br/>
<a NAME="E4:49_2"/><span class="kw">then </span>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="polyred.html#T26" target="_self" title="POLYRED:th.26">Th26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:49_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_2"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="polyred.html#D3" target="_self" title="POLYRED:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E40">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
