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<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</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">P</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">n</font>,<font color="Maroon">L</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">g</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>









<b>assume </b><a NAME="E1:85"/>
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font> <a href="polynom1.html#K25">-</a> <font color="Maroon">g</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">g1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:85"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">f1</font> <a href="polynom1.html#K25">-</a> <font color="Maroon">g1</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> &amp;  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font>,<font color="Maroon">f1</font> &amp;  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">g</font>,<font color="Maroon">g1</font> )
 <b>by </b><i/>;<br/>
<font color="Maroon">g1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="polynom1.html#K25">-</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">g1</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">g1</font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">g1</font>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">g1</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80">POLYNOM1:80</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f1</font>
<b>by </b><i><a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>
;<br/>
<b>hence </b><a NAME="E5:85"/>
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="rewrite1.html#R5">are_convergent_wrt</a>  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="rewrite1.html#D7">REWRITE1:def 7</a></i>;<br/>


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