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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#V2">admissible</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1_1_1"/><i><font color="Green">E14</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_1_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E5:14_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_1_1"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E6:14_1_1_1"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:14_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E5:14_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E9:14_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="groeb_1.html#E3:14_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E11:14_1_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><a NAME="E12:14_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="groeb_1.html#E11:14_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:14_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="polyred.html#R6">is_reducible_wrt</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E11:14_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="polyred.html#D8">POLYRED:def 8</a></i>;<br/><b>hence </b><a NAME="E14:14_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="rewrite1.html#R5">are_convergent_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5"><i><font color="Green">Lemma4</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:14_1_1"/>
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V9">locally-confluent</a>
 <b>by </b><i><a class="ref" href="rewrite1.html#D24">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">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom7.html#D2">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">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1_2_1"/><i><font color="Green">E13</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_1_2_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E5:14_1_2_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_2_1"><i><font color="Green">E14</font></i></a></i>;<br/><a NAME="E6:14_1_2_1"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:14_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E5:14_1_2_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><i><font color="Green">E16</font></i>: <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/><i><font color="Green">E17</font></i>: <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:14_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><a NAME="E11:14_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="groeb_1.html#E9:14_1_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E13:14_1_2_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><a NAME="E14:14_1_2_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E13:14_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:14_1_2_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E16:14_1_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><i><font color="Green">E21</font></i>: <font color="Maroon">c<sub>17</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E16:14_1_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><a NAME="E19:14_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="groeb_1.html#E9:14_1_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E21:14_1_2_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><a NAME="E22:14_1_2_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E21:14_1_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E23:14_1_2_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">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">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> or ( not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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">E25</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:14_1_2_1_4_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E14:14_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self">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">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></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">E26</font></i></a>, <a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>;<br/><a NAME="E5:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polyred.html#T2">POLYRED:2</a></i>;<br/><a NAME="E6:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T80">POLYNOM1:80</a></i>;<br/><a NAME="E7:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polyred.html#T3">POLYRED:3</a></i>;<br/><a NAME="E8:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E10:14_1_2_1_4_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E22:14_1_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E11:14_1_2_1_4_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></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">E27</font></i></a>, <a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>;<br/><a NAME="E12:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polyred.html#T6">POLYRED:6</a></i>;<br/><a NAME="E13:14_1_2_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T85">POLYNOM1:85</a></i>;<br/><a NAME="E14:14_1_2_1_4_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D23">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">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><a NAME="E2:14_1_2_1_4_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E14:14_1_2_1_4_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="polynom1.html#T83">POLYNOM1:83</a></i>;<br/><a NAME="E3:14_1_2_1_4_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T5">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">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>12</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>17</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1_4_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rewrite1.html#T13">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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>(  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:14_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="groeb_1.html#E18:14_1_2_1"><i><font color="Green">E21</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">E29</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D23">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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T80">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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T3">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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T2">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">E29</font></i></a>, <a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>20</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E4:14_1_2_1_4_1_1_1_1_2"/><i><font color="Green">E30</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>21</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E23:14_1_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="polyred.html#T45">POLYRED:45</a></i>;<br/><a NAME="E5:14_1_2_1_4_1_1_1_1_2"/>
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>12</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1_4_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rewrite1.html#T13">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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>(  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:14_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="groeb_1.html#E18:14_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="groeb_1.html#E14:14_1_2_1_4_1_1"><i><font color="Green">E28</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">E30</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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>(  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></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">E25</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:14_1_2_1_4_1_2"/><i><font color="Green">E26</font></i>: 
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E22:14_1_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self">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">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></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">E26</font></i></a>, <a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>;<br/><a NAME="E5:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polyred.html#T2">POLYRED:2</a></i>;<br/><a NAME="E6:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T80">POLYNOM1:80</a></i>;<br/><a NAME="E7:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polyred.html#T3">POLYRED:3</a></i>;<br/><a NAME="E8:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E10:14_1_2_1_4_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E14:14_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E11:14_1_2_1_4_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></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">E27</font></i></a>, <a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>;<br/><a NAME="E12:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polyred.html#T6">POLYRED:6</a></i>;<br/><a NAME="E13:14_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T85">POLYNOM1:85</a></i>;<br/><a NAME="E14:14_1_2_1_4_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D23">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">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></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">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><div class="add"><a NAME="E2:14_1_2_1_4_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E14:14_1_2_1_4_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="polynom1.html#T83">POLYNOM1:83</a></i>;<br/><a NAME="E3:14_1_2_1_4_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T5">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">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>12</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>17</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1_4_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rewrite1.html#T13">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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>(  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:14_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="groeb_1.html#E18:14_1_2_1"><i><font color="Green">E21</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">E29</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D23">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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T80">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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T3">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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T2">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">E29</font></i></a>, <a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>20</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E4:14_1_2_1_4_1_2_1_1_2"/><i><font color="Green">E30</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>21</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E23:14_1_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="polyred.html#T45">POLYRED:45</a></i>;<br/><a NAME="E5:14_1_2_1_4_1_2_1_1_2"/>
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>17</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:14_1_2_1_4_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rewrite1.html#T13">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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>(  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:14_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="groeb_1.html#E18:14_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="groeb_1.html#E14:14_1_2_1_4_1_2"><i><font color="Green">E28</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">E30</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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>(  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>17</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom7.html#D2">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">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></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">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>19</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T13">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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>(  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:14_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="groeb_1.html#E18:14_1_2_1"><i><font color="Green">E21</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">E25</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D23">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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T80">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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T3">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">c<sub>19</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T2">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">E25</font></i></a>, <a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>20</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:14_1_2_1_4_1_3_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E14:14_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self">Th1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E7:14_1_2_1_4_1_3_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E22:14_1_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self">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">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:14_1_2_1_4_1_3_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T80">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">c<sub>23</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T3">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">c<sub>23</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T2">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">E25</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">E26</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">E27</font></i></a>, <a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>23</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><font color="Maroon">c<sub>21</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K25">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K25">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></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">E26</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">E27</font></i></a>, <a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K25">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K25">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T1">POLYRED:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80">POLYNOM1:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K23">+</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80">POLYNOM1:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T3">POLYRED:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T2">POLYRED:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T6">POLYRED:6</a></i>
<br/>.= 
<font color="Maroon">c<sub>24</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="polynom1.html#T85">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">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>21</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E23:14_1_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="polyred.html#T45">POLYRED:45</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E13:14_1_2_1_4_1_3_1_1_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>25</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="ref" href="polyred.html#T49">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">c<sub>25</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>26</sub></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">E28</font></i></a>, <a class="ref" href="polynom1.html#D23">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">c<sub>25</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T80">POLYNOM1:80</a></i>;<br/><a NAME="E16:14_1_2_1_4_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T3">POLYRED:3</a></i>;<br/><a NAME="E17:14_1_2_1_4_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T2">POLYRED:2</a></i>;<br/><a NAME="E18:14_1_2_1_4_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T82">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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>(  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:14_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="groeb_1.html#E18:14_1_2_1"><i><font color="Green">E21</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">E28</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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>(  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E25:14_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="rewrite1.html#R5">are_convergent_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D7">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">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V9">locally-confluent</a>
 <b>by </b><i><a class="ref" href="rewrite1.html#D24">REWRITE1:def 24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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