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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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#V2" title="BAGORDER:attr.2">admissible</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#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</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  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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">p</font>, <font color="Maroon" title="c7">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>assume </b><a NAME="E1:95"/><i><font color="Green" title="E72">A1</font></i>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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; <font color="Maroon" title="c7">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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> )
 ;<br/>

<b>set </b><font color="Maroon" title="c8">PR</font> =  <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>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">f'</font> = <font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c10">h'</font> = <font color="Maroon" title="c7">h</font>, <font color="Maroon" title="c11">p'</font> = <font color="Maroon" title="c6">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>reconsider </b><font color="Maroon" title="c12">f'</font> = <font color="Maroon" title="c9">f'</font>, <font color="Maroon" title="c13">p'</font> = <font color="Maroon" title="c11">p'</font>, <font color="Maroon" title="c14">h'</font> = <font color="Maroon" title="c10">h'</font> as   <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>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">h'</font> = <font color="Maroon" title="c14">h'</font>, <font color="Maroon" title="c16">p'</font> = <font color="Maroon" title="c13">p'</font> as  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <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>by </b><i><a class="txt" href="polyred.html#E1:95"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/>
<a NAME="E5:95"/><i><font color="Green" title="E73">A2</font></i>: 
 <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="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</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>
 
<b>by </b><i><a class="txt" href="polyred.html#E1:95"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="polyred.html#T45" target="_self" title="POLYRED:th.45">Th45</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_1"/>
( <font color="Maroon" title="c12">f'</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> or <font color="Maroon" title="c12">f'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1"><b>case </b></a><a NAME="E1:95_1_1_1"/>
<font color="Maroon" title="c12">f'</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>
 ;<br/><div class="add"><a NAME="E2:95_1_1_1"/><b>then </b>
 <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="c12">f'</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font></span>)</span>,<font color="Maroon" title="c12">f'</font>
 <b>by </b><i><a class="txt" href="polyred.html#E5:95"><i><font color="Green" title="E73">A2</font></i></a>, <a class="ref" href="polyred.html#T2" target="_self" title="POLYRED:th.2">Th2</a></i>;<br/><a NAME="E3:95_1_1_1"/><b>then </b><i><font color="Green" title="E74">A3</font></i>: 
<font color="Maroon" title="c12">f'</font>,<font color="Maroon" title="c12">f'</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font></span>)</span> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_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="ref" href="rewrite1.html#T26" title="REWRITE1:th.26">REWRITE1:26</a></i>;<br/><a NAME="E4:95_1_1_1"/>
<font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>hence </b><a NAME="E5:95_1_1_1"/>
<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c5">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_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#E3:95_1_1_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_2"><b>case </b></a><a NAME="E1:95_1_1_2"/>
<font color="Maroon" title="c12">f'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c17">f'</font> = <font color="Maroon" title="c12">f'</font> as  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <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>by </b><i><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">f'</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font></span>)</span></span>)</span> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c17">f'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f'</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c17">f'</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f'</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="c17">f'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>
<br/>.= 
<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> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</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="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font>
<b>by </b><i><a class="ref" href="polyred.html#T2" target="_self" title="POLYRED:th.2">Th2</a></i>
;<br/><a NAME="E4:95_1_1_2"/><b>then </b><i><font color="Green" title="E74">A4</font></i>: 
 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f'</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font></span>)</span></span>)</span> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c17">f'</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>
 <b>by </b><i><a class="txt" href="polyred.html#E1:95"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="polyred.html#T45" target="_self" title="POLYRED:th.45">Th45</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_1_2_2_1"/>
( <font color="Maroon" title="c17">f'</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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> or <font color="Maroon" title="c17">f'</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font></span>)</span> <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:95_1_1_2_2_1_1"/>
<font color="Maroon" title="c17">f'</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c18">g'</font> = <font color="Maroon" title="c17">f'</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font></span>)</span> as  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <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>by </b><i><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E3:95_1_1_2_2_1_1"/><i><font color="Green" title="E75">A5</font></i>: 
<font color="Maroon" title="c18">g'</font>,<font color="Maroon" title="c17">f'</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_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#E4:95_1_1_2"><i><font color="Green" title="E74">A4</font></i></a>, <a class="ref" href="polyred.html#T51" target="_self" title="POLYRED:th.51">Th51</a></i>;<br/><a NAME="E4:95_1_1_2_2_1_1"/>
<font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E5:95_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c18">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>hence </b><a NAME="E6:95_1_1_2_2_1_1"/>
<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c5">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_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#E3:95_1_1_2_2_1_1"><i><font color="Green" title="E75">A5</font></i></a>, <a class="ref" href="rewrite1.html#T32" title="REWRITE1:th.32">REWRITE1:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:95_1_1_2_2_1_2"/><i><font color="Green" title="E75">A6</font></i>: 
<font color="Maroon" title="c17">f'</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font></span>)</span> <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>
 ;<br/><div class="add"><a NAME="E2:95_1_1_2_2_1_2"/>
<font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>then </b><i><font color="Green" title="E76">A7</font></i>: <font color="Maroon" title="c5">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font></span>)</span> = 
 <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>
<b>by </b><i><a class="txt" href="polyred.html#E1:95_1_1_2_2_1_2"><i><font color="Green" title="E75">A6</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/><b>then </b><i><font color="Green" title="E77">A8</font></i>: <font color="Maroon" title="c5">f</font> = 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T19" title="RLVECT_1:th.19">RLVECT_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">h</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_2_2_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_1_2_2_1_2_3"/><i><font color="Green" title="E78">A9</font></i>: 
 <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c15">h'</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>
 ;<br/><i><font color="Green" title="E79">A10</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">h'</font> = 
 <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_2_2_1_2_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:95_1_1_2_2_1_2_3_2"/>
<font color="Maroon" title="c18">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">h'</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c19">u'</font> = <font color="Maroon" title="c18">u</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c19">u'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c15">h'</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c19">u'</font>
<b>by </b><i><a class="ref" href="polynom1.html#D22" title="POLYNOM1:def.22">POLYNOM1:def 22</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="txt" href="polyred.html#E1:95_1_1_2_2_1_2_3"><i><font color="Green" title="E78">A9</font></i></a>, <a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>
;<br/><b>then </b><font color="Maroon" title="c15">h'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c19">u'</font> = 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T30" title="RLVECT_1:th.30">RLVECT_1:30</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></i>
<br/>.= 
<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> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c19">u'</font>
<b>by </b><i><a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>
;<br/><b>hence </b><a NAME="E5:95_1_1_2_2_1_2_3_2"/>
<font color="Maroon" title="c15">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">u</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:95_1_1_2_2_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="polyred.html#E1:95"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="polyred.html#E2:95_1_1_2_2_1_2_3"><i><font color="Green" title="E79">A10</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c18">mh'</font> =  <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c15">h'</font> as  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <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>by </b><i><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">x</font> = <font color="Maroon" title="c18">mh'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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/><font color="Maroon" title="c7">h</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x</font> = 
<font color="Maroon" title="c18">mh'</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c15">h'</font>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
<br/>.= 
 <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>
<b>by </b><i><a class="ref" href="polyred.html#T3" target="_self" title="POLYRED:th.3">Th3</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/><a NAME="E10:95_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E78">A11</font></i>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">mh'</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" title="RLVECT_1:th.19">RLVECT_1:19</a></i>;<br/><a NAME="E11:95_1_1_2_2_1_2"/>
 <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="c18">mh'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</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>
 <b>by </b><i><a class="txt" href="polyred.html#E1:95"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="polyred.html#T45" target="_self" title="POLYRED:th.45">Th45</a></i>;<br/><a NAME="E12:95_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E79">A12</font></i>: 
<font color="Maroon" title="c18">mh'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</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> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_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="ref" href="rewrite1.html#T26" title="REWRITE1:th.26">REWRITE1:26</a></i>;<br/><a NAME="E13:95_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">h</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">mh'</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c16">p'</font>
 <b>by </b><i><a class="txt" href="polyred.html#E10:95_1_1_2_2_1_2"><i><font color="Green" title="E78">A11</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>hence </b><a NAME="E14:95_1_1_2_2_1_2"/>
<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c5">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_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#E3:95_1_1_2_2_1_2"><i><font color="Green" title="E76">A7</font></i></a>, <a class="txt" href="polyred.html#E4:95_1_1_2_2_1_2"><i><font color="Green" title="E77">A8</font></i></a>, <a class="txt" href="polyred.html#E12:95_1_1_2_2_1_2"><i><font color="Green" title="E79">A12</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:95_1_1_2"/>
<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c5">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_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>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:95"/>
<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c5">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_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>
 ;<br/>


</div>
