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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#V2">admissible</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <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="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <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>assume </b><a NAME="E1:23"/><i><font color="Green">E40</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> is <a href="rewrite1.html#V8">with_Church-Rosser_property</a>
 ;<br/>

<b>set </b><font color="Maroon">R</font> =  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><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>;<br/><b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 ;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</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> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f'</font> as   <a href="struct_0.html#NM1">Element</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>reconsider </b><font color="Maroon">e</font> =  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> as   <a href="struct_0.html#NM1">Element</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> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="struct_0.html#NM1">Element</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/><a NAME="E6:23_1"/>
<font color="Maroon">f</font> <a href="polynom1.html#K25">-</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> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/><a NAME="E7:23_1"/><b>then </b>
<font color="Maroon">f'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i>, <a class="ref" href="polyred.html#T4">POLYRED:4</a></i>;<br/><a NAME="E8:23_1"/><b>then </b>
<font color="Maroon">f'</font>,<font color="Maroon">e</font> <a href="polyred.html#R11">are_congruent_mod</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="polyred.html#D14">POLYRED:def 14</a></i>;<br/><a NAME="E9:23_1"/><b>then </b>
<font color="Maroon">f'</font>,<font color="Maroon">e</font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#T58">POLYRED:58</a></i>;<br/><a NAME="E10:23_1"/><b>then </b>
<font color="Maroon">f'</font>,<font color="Maroon">e</font> <a href="rewrite1.html#R5">are_convergent_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="rewrite1.html#D23">REWRITE1:def 23</a></i>;<br/><b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:23_1"/><i><font color="Green">E43</font></i>: 
(  <a href="polyred.html#K3">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">c</font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>,<font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D7">REWRITE1:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1"/>
<font color="Maroon">c'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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">h</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="E3:23_1_1"/><i><font color="Green">E44</font></i>: 
(  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">h</font>,<font color="Maroon">P</font>,<font color="Maroon">T</font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">h</font>,<font color="Maroon">c'</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">pp</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="E5:23_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">pp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp;  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">h</font>,<font color="Maroon">pp</font>,<font color="Maroon">T</font> )
 <b>by </b><i>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><a NAME="E6:23_1_1"/>
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="polyred.html#R6">is_reducible_wrt</a> <font color="Maroon">pp</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="polyred.html#D8">POLYRED:def 8</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="polyred.html#T37">POLYRED:37</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:23_1"/>
 <a href="polyred.html#K3">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#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:23"/>
for <font color="Olive">f</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/> <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">f</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/>


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