<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</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 <a href="xboole_0.html#K1">{}</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</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>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ideal_1.html#V1">add-closed</a> <a href="ideal_1.html#V2">left-ideal</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:44"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E34</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 </span>}</span> ;<br/><a NAME="E3:44_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:44_1"><i><font color="Green">E34</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:44_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 ;<br/><b>thus </b><a NAME="E6:44_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:44_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="ideal_1.html#V1">add-closed</a> <a href="ideal_1.html#V2">left-ideal</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <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>5</sub></font> ;<br/>
<a NAME="E7:44"/><i><font color="Green">E34</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> holds not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:44_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:44_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:44_2"><i><font color="Green">E35</font></i></a></i>;<br/><b>hence </b><a NAME="E3:44_2_1"/>
<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>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></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><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:44_2_2"/>
<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>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 ;<br/><a NAME="E2:44_2_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <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="44_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_2_2_1"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:44_2_2_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:44_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="groeb_1.html#E2:44_2_2"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E3:44_2_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:44_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:44_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:44"><i><font color="Green">E33</font></i></a>, <a class="txt" href="groeb_1.html#E2:44_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:44_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E3:44_3"/><i><font color="Green">E35</font></i>: 
<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>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E7:44"><i><font color="Green">E34</font></i></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>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <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>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E3:44_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E6:44_3"/><i><font color="Green">E36</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="polynom7.html#T3">POLYNOM7:3</a></i>;<br/><a NAME="E7:44_3"/>
<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>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E8:44_3"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom7.html#T1">POLYNOM7:1</a></i>;<br/><a NAME="E9:44_3"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E10:44_3"/><b>then </b>
 <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="polynom7.html#T3">POLYNOM7:3</a></i>;<br/><a NAME="E11:44_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="polyred.html#R6">is_reducible_wrt</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E6:44_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="polyred.html#T36">POLYRED:36</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>5</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E13:44_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D8">POLYRED:def 8</a></i>;<br/><a NAME="E14:44_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E13:44_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><b>hence </b><a NAME="E15:44_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D9">POLYRED:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:44"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E1:44"><i><font color="Green">E33</font></i></a>, <a class="ref" href="groeb_1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E10:44"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>5</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#R3">divides</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="groeb_1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E11:44"/><b>then </b>
 <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="groeb_1.html#K3" target="_self">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T28" target="_self">Th28</a></i>;<br/>
<b>hence </b><a NAME="E12:44"/>
<font color="Maroon">c<sub>4</sub></font> <a href="groeb_1.html#R2" target="_self">is_Groebner_basis_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:44"><i><font color="Green">E33</font></i></a>, <a class="ref" href="groeb_1.html#T29" target="_self">Th29</a></i>;<br/>


</div>
