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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  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:15"/>
ex <font color="Olive">b<sub>1</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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span> <a href="ideal_1.html#K7">-Ideal</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</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="E3:15"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span> <a href="ideal_1.html#K7">-Ideal</a>  )
 ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_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="E1:15_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>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> <font color="Maroon">c<sub>4</sub></font>,<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> <font color="Maroon">c<sub>4</sub></font>,<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="E3:15_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> <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#E3:15_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><i><font color="Green">E16</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#E3:15_1"><i><font color="Green">E15</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/><b>consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:15_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>13</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>14</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>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:15_1"><i><font color="Green">E14</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>15</sub></font> = <font color="Maroon">c<sub>14</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#E7:15_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><i><font color="Green">E18</font></i>: <font color="Maroon">c<sub>15</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#E7:15_1"><i><font color="Green">E17</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="E10:15_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:15_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="rewrite1.html#T30">REWRITE1:30</a></i>;<br/><a NAME="E11:15_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T32">REWRITE1:32</a></i>;<br/><a NAME="E12:15_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:15_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="groeb_1.html#E9:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="groeb_1.html#E10:15_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="rewrite1.html#T31">REWRITE1:31</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/><a NAME="E15:15_1"/>
<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="polyred.html#R11">are_congruent_mod</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="groeb_1.html#E12:15_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="polyred.html#T57">POLYRED:57</a></i>;<br/><a NAME="E16:15_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="rewrite1.html#R2">are_convertible_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>5</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="polyred.html#T58">POLYRED:58</a></i>;<br/><b>set </b><font color="Maroon">c<sub>20</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>5</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</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>5</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> as  <a href="rewrite1.html#V3">strongly-normalizing</a> <a href="rewrite1.html#V9">locally-confluent</a> <a href="relat_1.html#NM1">Relation</a> <b>by </b><i><a class="ref" href="groeb_1.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E18:15_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> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:15_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="groeb_1.html#E9:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="groeb_1.html#E16:15_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rewrite1.html#D23">REWRITE1:def 23</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:15_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D7">REWRITE1:def 7</a></i>;<br/><a NAME="E21:15_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</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>5</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:15"><i><font color="Green">E13</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E22:15_1"/><b>then </b>
<span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E23:15_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="groeb_1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E24:15_1"/><b>then </b>
(  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<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="Maroon">c<sub>22</sub></font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<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="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E20:15_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rewrite1.html#T23">REWRITE1:23</a></i>;<br/><b>hence </b><a NAME="E25:15_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> <font color="Maroon">c<sub>4</sub></font>,<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="E5:15"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<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/>


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