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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#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="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>set </b><font color="Maroon" title="c5">R</font> =  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>suppose </b></a><a NAME="E1:14_1_1"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1_1_1"/><i><font color="Green" title="E14">A2</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c9">p'</font>, <font color="Maroon" title="c10">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_1_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c9">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<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></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">p'</font>,<font color="Maroon" title="c10">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c10">q</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="txt" href="groeb_1.html#E3:14_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E5:14_1_1_1"/>
not <font color="Maroon" title="c9">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<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></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:14_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">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>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">p'</font> = <font color="Maroon" title="c9">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="groeb_1.html#E3:14_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E8:14_1_1_1"/>
<font color="Maroon" title="c12">p'</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c11">q</font>,<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="groeb_1.html#E3:14_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="polyred.html#D13" title="POLYRED:def.13">POLYRED:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">g</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="E10:14_1_1_1"/><i><font color="Green" title="E16">A4</font></i>: 
( <font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c12">p'</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></i>;<br/><a NAME="E11:14_1_1_1"/>
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="groeb_1.html#E10:14_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E12:14_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">p'</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</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>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:14_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="polyred.html#D8" title="POLYRED:def.8">POLYRED:def 8</a></i>;<br/><b>hence </b><a NAME="E13:14_1_1_1"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</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> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:14_1_1"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> is <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>
 <b>by </b><i><a class="ref" href="rewrite1.html#D24" title="REWRITE1:def.24">REWRITE1:def 24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><b>suppose </b></a><a NAME="E1:14_1_2"/>
<font color="Maroon" title="c4">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>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c4">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="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1_2_1"/><i><font color="Green" title="E13">A5</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c10">pa</font>, <font color="Maroon" title="c11">pb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_1_2_1"/><i><font color="Green" title="E14">A6</font></i>: 
( <font color="Maroon" title="c10">pa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<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></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c11">pb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">pa</font>,<font color="Maroon" title="c11">pb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">pb</font> = <font color="Maroon" title="c11">pb</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="txt" href="groeb_1.html#E3:14_1_2_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E5:14_1_2_1"/>
not <font color="Maroon" title="c10">pa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<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></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_2_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:14_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">pa</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>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">pa</font> = <font color="Maroon" title="c10">pa</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="groeb_1.html#E3:14_1_2_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><i><font color="Green" title="E15">A7</font></i>: <font color="Maroon" title="c13">pa</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_2_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c7">a</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/><i><font color="Green" title="E16">A8</font></i>: <font color="Maroon" title="c12">pb</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_2_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/><a NAME="E10:14_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">pa</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c12">pb</font>,<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="groeb_1.html#E8:14_1_2_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="polyred.html#D13" title="POLYRED:def.13">POLYRED:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">p'</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="E12:14_1_2_1"/><i><font color="Green" title="E17">A9</font></i>: 
( <font color="Maroon" title="c14">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c13">pa</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c12">pb</font>,<font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></i>;<br/><a NAME="E13:14_1_2_1"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c13">pa</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c12">pb</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E12:14_1_2_1"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">pa'</font>, <font color="Maroon" title="c16">pc</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:14_1_2_1"/><i><font color="Green" title="E19">A11</font></i>: 
( <font color="Maroon" title="c15">pa'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<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></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c16">pc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">pa'</font>,<font color="Maroon" title="c16">pc</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">pc</font> = <font color="Maroon" title="c16">pc</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="txt" href="groeb_1.html#E15:14_1_2_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><i><font color="Green" title="E20">A12</font></i>: <font color="Maroon" title="c17">pc</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E15:14_1_2_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c9">c</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/><a NAME="E18:14_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">pa</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c17">pc</font>,<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="groeb_1.html#E8:14_1_2_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="polyred.html#D13" title="POLYRED:def.13">POLYRED:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">p'</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="E20:14_1_2_1"/><i><font color="Green" title="E21">A13</font></i>: 
( <font color="Maroon" title="c18">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c13">pa</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c17">pc</font>,<font color="Maroon" title="c18">p'</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></i>;<br/><a NAME="E21:14_1_2_1"/><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c13">pa</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c17">pc</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E20:14_1_2_1"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E22:14_1_2_1"/><i><font color="Green" title="E23">A15</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_2_1_4_1"/>
( <font color="Maroon" title="c12">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 ( not <font color="Maroon" title="c12">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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; not <font color="Maroon" title="c17">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="14_1_2_1_4_1_1"><b>case </b></a><a NAME="E1:14_1_2_1_4_1_1"/><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon" title="c12">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 consider </b><font color="Maroon" title="c19">mb</font> being   <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E3:14_1_2_1_4_1_1"/><i><font color="Green" title="E25">A17</font></i>: 
 <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">pa</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E13:14_1_2_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self" title="GROEB_1:th.1">Th1</a></i>;<br/><a NAME="E4:14_1_2_1_4_1_1"/>
<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="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">pa</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> <span class="p3">(<span class="default"><font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_2_1_4_1_1"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>;<br/><a NAME="E5:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">pa</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> <span class="p3">(<span class="default"><font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></i>;<br/><a NAME="E6:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">pa</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> <span class="p3">(<span class="default"><font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</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/><a NAME="E7:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">pa</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" title="POLYRED:th.3">POLYRED:3</a></i>;<br/><a NAME="E8:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">pa</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">mc</font> being   <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E10:14_1_2_1_4_1_1"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c17">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E21:14_1_2_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self" title="GROEB_1:th.1">Th1</a></i>;<br/><a NAME="E11:14_1_2_1_4_1_1"/>
<font color="Maroon" title="c17">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></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> <span class="p2">(<span class="default"><font color="Maroon" title="c20">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:14_1_2_1_4_1_1"><i><font color="Green" title="E26">A18</font></i></a>, <a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>;<br/><a NAME="E12:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c17">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span> <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="c20">mc</font></span>)</span> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="polyred.html#T6" title="POLYRED:th.6">POLYRED:6</a></i>;<br/><a NAME="E13:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c17">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mb</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="c20">mc</font></span>)</span></span>)</span> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T85" title="POLYNOM1:th.85">POLYNOM1:85</a></i>;<br/><a NAME="E14:14_1_2_1_4_1_1"/><b>then </b><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c17">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c20">mc</font></span>)</span> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_4_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_2_1_4_1_1_1_1"/>
( <font color="Maroon" title="c19">mb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">mc</font> or <font color="Maroon" title="c19">mb</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">mc</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1_1_1_1_1"><b>case </b></a><a NAME="E1:14_1_2_1_4_1_1_1_1_1"/>
<font color="Maroon" title="c19">mb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">mc</font>
 ;<br/><div class="add"><a NAME="E2:14_1_2_1_4_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">pc</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="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E14:14_1_2_1_4_1_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="ref" href="polynom1.html#T83" title="POLYNOM1:th.83">POLYNOM1:83</a></i>;<br/><a NAME="E3:14_1_2_1_4_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 <b>by </b><i><a class="ref" href="polyred.html#T5" title="POLYRED:th.5">POLYRED:5</a></i>;<br/><a NAME="E4:14_1_2_1_4_1_1_1_1_1"/><b>then </b>
(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c12">pb</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> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c17">pc</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="groeb_1.html#E1:14_1_2_1_4_1_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="rewrite1.html#T13" title="REWRITE1:th.13">REWRITE1:13</a></i>;<br/><b>hence </b><a NAME="E5:14_1_2_1_4_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">b</font>,<font color="Olive" title="b1">d</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">c</font>,<font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:14_1_2_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="groeb_1.html#E17:14_1_2_1"><i><font color="Green" title="E20">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1_1_1_1_2"><b>case </b></a><a NAME="E1:14_1_2_1_4_1_1_1_1_2"/><i><font color="Green" title="E28">A20</font></i>: 
<font color="Maroon" title="c19">mb</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">mc</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_4_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_2_1_4_1_1_1_1_2_1"/>
<font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c20">mc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/><a NAME="E2:14_1_2_1_4_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">mb</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="c20">mc</font></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">mc</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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">mc</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>;<br/><a NAME="E3:14_1_2_1_4_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">mb</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="c20">mc</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">mc</font></span>)</span> <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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">mc</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>;<br/><a NAME="E4:14_1_2_1_4_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">mb</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> <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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">mc</font>
 <b>by </b><i><a class="ref" href="polyred.html#T3" title="POLYRED:th.3">POLYRED:3</a></i>;<br/><a NAME="E5:14_1_2_1_4_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">mb</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">mc</font>
 <b>by </b><i><a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></i>;<br/><b>hence </b><a NAME="E6:14_1_2_1_4_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1_4_1_1_1_1_2"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c21">hh</font> = <font color="Maroon" title="c19">mb</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c20">mc</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/><a NAME="E4:14_1_2_1_4_1_1_1_1_2"/><i><font color="Green" title="E29">A21</font></i>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c21">hh</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">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="groeb_1.html#E22:14_1_2_1"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="polyred.html#T45" title="POLYRED:th.45">POLYRED:45</a></i>;<br/><a NAME="E5:14_1_2_1_4_1_1_1_1_2"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c12">pb</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="groeb_1.html#E1:14_1_2_1_4_1_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="rewrite1.html#T13" title="REWRITE1:th.13">REWRITE1:13</a></i>;<br/><b>hence </b><a NAME="E6:14_1_2_1_4_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">b</font>,<font color="Olive" title="b1">d</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">c</font>,<font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:14_1_2_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="groeb_1.html#E17:14_1_2_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="groeb_1.html#E14:14_1_2_1_4_1_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="groeb_1.html#E4:14_1_2_1_4_1_1_1_1_2"><i><font color="Green" title="E29">A21</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:14_1_2_1_4_1_1"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">b</font>,<font color="Olive" title="b1">d</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">c</font>,<font color="Olive" title="b1">d</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1_2"><b>case </b></a><a NAME="E1:14_1_2_1_4_1_2"/><i><font color="Green" title="E24">A22</font></i>: 
<font color="Maroon" title="c17">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 consider </b><font color="Maroon" title="c19">mc</font> being   <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E3:14_1_2_1_4_1_2"/><i><font color="Green" title="E25">A23</font></i>: 
 <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">pa</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E21:14_1_2_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self" title="GROEB_1:th.1">Th1</a></i>;<br/><a NAME="E4:14_1_2_1_4_1_2"/>
<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="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">pa</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> <span class="p3">(<span class="default"><font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_2_1_4_1_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>;<br/><a NAME="E5:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">pa</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> <span class="p3">(<span class="default"><font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></i>;<br/><a NAME="E6:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">pa</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> <span class="p3">(<span class="default"><font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</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/><a NAME="E7:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">pa</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" title="POLYRED:th.3">POLYRED:3</a></i>;<br/><a NAME="E8:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">pa</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">mb</font> being   <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E10:14_1_2_1_4_1_2"/><i><font color="Green" title="E26">A24</font></i>: 
<font color="Maroon" title="c12">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E13:14_1_2_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self" title="GROEB_1:th.1">Th1</a></i>;<br/><a NAME="E11:14_1_2_1_4_1_2"/>
<font color="Maroon" title="c12">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></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> <span class="p2">(<span class="default"><font color="Maroon" title="c20">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:14_1_2_1_4_1_2"><i><font color="Green" title="E26">A24</font></i></a>, <a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>;<br/><a NAME="E12:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c12">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span> <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="c20">mb</font></span>)</span> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="polyred.html#T6" title="POLYRED:th.6">POLYRED:6</a></i>;<br/><a NAME="E13:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c12">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mc</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="c20">mb</font></span>)</span></span>)</span> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T85" title="POLYNOM1:th.85">POLYNOM1:85</a></i>;<br/><a NAME="E14:14_1_2_1_4_1_2"/><b>then </b><i><font color="Green" title="E27">A25</font></i>: 
<font color="Maroon" title="c12">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c20">mb</font></span>)</span> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_4_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_2_1_4_1_2_1_1"/>
( <font color="Maroon" title="c20">mb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">mc</font> or <font color="Maroon" title="c20">mb</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">mc</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1_2_1_1_1"><b>case </b></a><a NAME="E1:14_1_2_1_4_1_2_1_1_1"/>
<font color="Maroon" title="c20">mb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">mc</font>
 ;<br/><div class="add"><a NAME="E2:14_1_2_1_4_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">pb</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="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E14:14_1_2_1_4_1_2"><i><font color="Green" title="E27">A25</font></i></a>, <a class="ref" href="polynom1.html#T83" title="POLYNOM1:th.83">POLYNOM1:83</a></i>;<br/><a NAME="E3:14_1_2_1_4_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 <b>by </b><i><a class="ref" href="polyred.html#T5" title="POLYRED:th.5">POLYRED:5</a></i>;<br/><a NAME="E4:14_1_2_1_4_1_2_1_1_1"/><b>then </b>
(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c12">pb</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> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c17">pc</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="groeb_1.html#E1:14_1_2_1_4_1_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="rewrite1.html#T13" title="REWRITE1:th.13">REWRITE1:13</a></i>;<br/><b>hence </b><a NAME="E5:14_1_2_1_4_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">b</font>,<font color="Olive" title="b1">d</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">c</font>,<font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:14_1_2_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="groeb_1.html#E17:14_1_2_1"><i><font color="Green" title="E20">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1_2_1_1_2"><b>case </b></a><a NAME="E1:14_1_2_1_4_1_2_1_1_2"/><i><font color="Green" title="E28">A26</font></i>: 
<font color="Maroon" title="c20">mb</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">mc</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_4_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_2_1_4_1_2_1_1_2_1"/>
<font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c20">mb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/><a NAME="E2:14_1_2_1_4_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">mc</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="c20">mb</font></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">mb</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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">mb</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>;<br/><a NAME="E3:14_1_2_1_4_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">mc</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="c20">mb</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">mb</font></span>)</span> <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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">mb</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>;<br/><a NAME="E4:14_1_2_1_4_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">mc</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> <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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">mb</font>
 <b>by </b><i><a class="ref" href="polyred.html#T3" title="POLYRED:th.3">POLYRED:3</a></i>;<br/><a NAME="E5:14_1_2_1_4_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">mc</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">mb</font>
 <b>by </b><i><a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></i>;<br/><b>hence </b><a NAME="E6:14_1_2_1_4_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1_4_1_2_1_1_2"><i><font color="Green" title="E28">A26</font></i></a>, <a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c21">hh</font> = <font color="Maroon" title="c19">mc</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c20">mb</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/><a NAME="E4:14_1_2_1_4_1_2_1_1_2"/><i><font color="Green" title="E29">A27</font></i>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c21">hh</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">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="groeb_1.html#E22:14_1_2_1"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="polyred.html#T45" title="POLYRED:th.45">POLYRED:45</a></i>;<br/><a NAME="E5:14_1_2_1_4_1_2_1_1_2"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c17">pc</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="groeb_1.html#E1:14_1_2_1_4_1_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="rewrite1.html#T13" title="REWRITE1:th.13">REWRITE1:13</a></i>;<br/><b>hence </b><a NAME="E6:14_1_2_1_4_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">b</font>,<font color="Olive" title="b1">d</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">c</font>,<font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:14_1_2_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="groeb_1.html#E17:14_1_2_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="groeb_1.html#E14:14_1_2_1_4_1_2"><i><font color="Green" title="E27">A25</font></i></a>, <a class="txt" href="groeb_1.html#E4:14_1_2_1_4_1_2_1_1_2"><i><font color="Green" title="E29">A27</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:14_1_2_1_4_1_2"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">b</font>,<font color="Olive" title="b1">d</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">c</font>,<font color="Olive" title="b1">d</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1_3"><b>case </b></a><a NAME="E1:14_1_2_1_4_1_3"/>
( not <font color="Maroon" title="c12">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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; not <font color="Maroon" title="c17">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c19">pb</font> = <font color="Maroon" title="c12">pb</font>, <font color="Maroon" title="c20">pc</font> = <font color="Maroon" title="c17">pc</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_4_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_2_1_4_1_3_1_1"/>
( <font color="Maroon" title="c19">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">pc</font> or <font color="Maroon" title="c19">pb</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">pc</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1_3_1_1_1"><b>case </b></a><a NAME="E1:14_1_2_1_4_1_3_1_1_1"/>
<font color="Maroon" title="c19">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">pc</font>
 ;<br/><div class="add"><a NAME="E2:14_1_2_1_4_1_3_1_1_1"/><b>then </b>
(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c19">pb</font>,<font color="Maroon" title="c19">pb</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c20">pc</font>,<font color="Maroon" title="c19">pb</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T13" title="REWRITE1:th.13">REWRITE1:13</a></i>;<br/><b>hence </b><a NAME="E3:14_1_2_1_4_1_3_1_1_1"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">b</font>,<font color="Olive" title="b1">d</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">c</font>,<font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:14_1_2_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="groeb_1.html#E17:14_1_2_1"><i><font color="Green" title="E20">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_4_1_3_1_1_2"><b>case </b></a><a NAME="E1:14_1_2_1_4_1_3_1_1_2"/><i><font color="Green" title="E24">A28</font></i>: 
<font color="Maroon" title="c19">pb</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">pc</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_4_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_2_1_4_1_3_1_1_2_1"/>
<font color="Maroon" title="c19">pb</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c20">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/><a NAME="E2:14_1_2_1_4_1_3_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">pb</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="c20">pc</font></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">pc</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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">pc</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>;<br/><a NAME="E3:14_1_2_1_4_1_3_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">pb</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="c20">pc</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">pc</font></span>)</span> <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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">pc</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>;<br/><a NAME="E4:14_1_2_1_4_1_3_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">pb</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> <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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c20">pc</font>
 <b>by </b><i><a class="ref" href="polyred.html#T3" title="POLYRED:th.3">POLYRED:3</a></i>;<br/><a NAME="E5:14_1_2_1_4_1_3_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">pb</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">pc</font>
 <b>by </b><i><a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></i>;<br/><b>hence </b><a NAME="E6:14_1_2_1_4_1_3_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1_4_1_3_1_1_2"><i><font color="Green" title="E24">A28</font></i></a>, <a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c21">h</font> = <font color="Maroon" title="c19">pb</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c20">pc</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>consider </b><font color="Maroon" title="c22">mb</font> being   <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E5:14_1_2_1_4_1_3_1_1_2"/><i><font color="Green" title="E25">A29</font></i>: 
<font color="Maroon" title="c19">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">pa</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E13:14_1_2_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self" title="GROEB_1:th.1">Th1</a></i>;<br/><b>consider </b><font color="Maroon" title="c23">mc</font> being   <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E7:14_1_2_1_4_1_3_1_1_2"/><i><font color="Green" title="E26">A30</font></i>: 
<font color="Maroon" title="c20">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">pa</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E21:14_1_2_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self" title="GROEB_1:th.1">Th1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_4_1_3_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_2_1_4_1_3_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c22">mb</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c23">mc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/><a NAME="E2:14_1_2_1_4_1_3_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c23">mc</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="c22">mb</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c22">mb</font></span>)</span> <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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c22">mb</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>;<br/><a NAME="E3:14_1_2_1_4_1_3_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c23">mc</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> <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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c22">mb</font>
 <b>by </b><i><a class="ref" href="polyred.html#T3" title="POLYRED:th.3">POLYRED:3</a></i>;<br/><a NAME="E4:14_1_2_1_4_1_3_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c23">mc</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">mb</font>
 <b>by </b><i><a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></i>;<br/><b>hence </b><a NAME="E5:14_1_2_1_4_1_3_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1_4_1_3_1_1_2"><i><font color="Green" title="E24">A28</font></i></a>, <a class="txt" href="groeb_1.html#E5:14_1_2_1_4_1_3_1_1_2"><i><font color="Green" title="E25">A29</font></i></a>, <a class="txt" href="groeb_1.html#E7:14_1_2_1_4_1_3_1_1_2"><i><font color="Green" title="E26">A30</font></i></a>, <a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c24">hh</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c22">mb</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c23">mc</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/><font color="Maroon" title="c21">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">pa</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">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> <span class="p2">(<span class="default"><font color="Maroon" title="c13">pa</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="groeb_1.html#E5:14_1_2_1_4_1_3_1_1_2"><i><font color="Green" title="E25">A29</font></i></a>, <a class="txt" href="groeb_1.html#E7:14_1_2_1_4_1_3_1_1_2"><i><font color="Green" title="E26">A30</font></i></a>, <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="c13">pa</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">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> <span class="p2">(<span class="default"><font color="Maroon" title="c13">pa</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span></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="c13">pa</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <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="c13">pa</font></span>)</span> <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> <span class="p3">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T1" title="POLYRED:th.1">POLYRED:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">pa</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> <span class="p3">(<span class="default"><font color="Maroon" title="c22">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span> <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="c13">pa</font></span>)</span> <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> <span class="p3">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon" title="c13">pa</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span> <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="c13">pa</font></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></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"><span class="p2">(<span class="default"><font color="Maroon" title="c13">pa</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c13">pa</font></span>)</span></span>)</span> <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> <span class="p3">(<span class="default"><font color="Maroon" title="c22">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></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"><span class="p2">(<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="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T3" title="POLYRED:th.3">POLYRED:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">mb</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></i>
<br/>.= 
<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="c22">mb</font></span>)</span> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">mc</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T6" title="POLYRED:th.6">POLYRED:6</a></i>
<br/>.= 
<font color="Maroon" title="c24">hh</font> <a href="polynom1.html#K29" title="POLYNOM1:func.29">*'</a> <font color="Maroon" title="c6">p</font>
<b>by </b><i><a class="ref" href="polynom1.html#T85" title="POLYNOM1:th.85">POLYNOM1:85</a></i>
;<br/><a NAME="E11:14_1_2_1_4_1_3_1_1_2"/><b>then </b>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c21">h</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="groeb_1.html#E22:14_1_2_1"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="polyred.html#T45" title="POLYRED:th.45">POLYRED:45</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">f1</font>, <font color="Maroon" title="c26">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="E13:14_1_2_1_4_1_3_1_1_2"/><i><font color="Green" title="E27">A31</font></i>: 
( <font color="Maroon" title="c25">f1</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c26">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c19">pb</font>,<font color="Maroon" title="c25">f1</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c20">pc</font>,<font color="Maroon" title="c26">g1</font> )
 <b>by </b><i><a class="ref" href="polyred.html#T49" title="POLYRED:th.49">POLYRED:49</a></i>;<br/><a NAME="E14:14_1_2_1_4_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c25">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="c26">g1</font></span>)</span></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c26">g1</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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c26">g1</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E13:14_1_2_1_4_1_3_1_1_2"><i><font color="Green" title="E27">A31</font></i></a>, <a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>;<br/><a NAME="E15:14_1_2_1_4_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c25">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="c26">g1</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c26">g1</font></span>)</span> <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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c26">g1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>;<br/><a NAME="E16:14_1_2_1_4_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c25">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> <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="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c26">g1</font>
 <b>by </b><i><a class="ref" href="polyred.html#T3" title="POLYRED:th.3">POLYRED:3</a></i>;<br/><a NAME="E17:14_1_2_1_4_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c25">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">g1</font>
 <b>by </b><i><a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></i>;<br/><a NAME="E18:14_1_2_1_4_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c25">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">g1</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="E19:14_1_2_1_4_1_3_1_1_2"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">b</font>,<font color="Olive" title="b1">d</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">c</font>,<font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:14_1_2_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="groeb_1.html#E17:14_1_2_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="groeb_1.html#E13:14_1_2_1_4_1_3_1_1_2"><i><font color="Green" title="E27">A31</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:14_1_2_1_4_1_3"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">b</font>,<font color="Olive" title="b1">d</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">c</font>,<font color="Olive" title="b1">d</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E24:14_1_2_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</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> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D7" title="REWRITE1:def.7">REWRITE1:def 7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:14_1_2"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> is <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>
 <b>by </b><i><a class="ref" href="rewrite1.html#D24" title="REWRITE1:def.24">REWRITE1:def 24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
