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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>









<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:86"/>
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>6</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:86"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="rewrite1.html#R5">are_convergent_wrt</a>  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="polyred.html#T50" target="_self">Th50</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:86"/><i><font color="Green">E66</font></i>: 
(  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp;  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D7">REWRITE1:def 7</a></i>;<br/>
<a NAME="E5:86"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a>  <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="polyred.html#E4:86"><i><font color="Green">E66</font></i></a>, <a class="ref" href="rewrite1.html#T25">REWRITE1:25</a></i>;<br/>
<a NAME="E6:86"/>
(  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:86"/><b>then </b>
( <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="polyred.html#E4:86"><i><font color="Green">E66</font></i></a>, <a class="txt" href="polyred.html#E5:86"><i><font color="Green">E67</font></i></a>, <a class="ref" href="rewrite1.html#T23">REWRITE1:23</a></i>;<br/>
<a NAME="E8:86"/><b>then </b>
<span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#T17">REWRITE1:17</a></i>;<br/>
<b>hence </b><a NAME="E9:86"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D4">REWRITE1:def 4</a></i>;<br/>


</div>
