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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>;<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>;<br/><b>let </b><font color="Maroon" title="c3">L</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="vectsp_1.html#NV11" title="VECTSP_1:NV.11">Field-like</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</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>;<br/>









<b>assume </b><a NAME="E1:85"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">f</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c6">g</font>, <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">f1</font>, <font color="Maroon" title="c8">g1</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><b> such that </b><br/><a NAME="E3:85"/><i><font color="Green" title="E65">A1</font></i>: 
( <font color="Maroon" title="c7">f1</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c8">g1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">f1</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c8">g1</font> )
 <b>by </b><i><a class="ref" href="polyred.html#T49" target="_self" title="POLYRED:th.49">Th49</a></i>;<br/>
<font color="Maroon" title="c8">g1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c8">g1</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c8">g1</font>
<b>by </b><i><a class="txt" href="polyred.html#E3:85"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="polyred.html#T2" target="_self" title="POLYRED:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c8">g1</font></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c8">g1</font>
<b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>
<br/>.= 
<font color="Maroon" title="c7">f1</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c8">g1</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c8">g1</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>
<br/>.= 
<font color="Maroon" title="c7">f1</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T3" target="_self" title="POLYRED:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c7">f1</font>
<b>by </b><i><a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></i>
;<br/>
<b>hence </b><a NAME="E5:85"/>
<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="polyred.html#E3:85"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="rewrite1.html#D7" title="REWRITE1:def.7">REWRITE1:def 7</a></i>;<br/>


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