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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</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#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V6">right_unital</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</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>, <font color="Maroon">c<sub>5</sub></font> be  <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>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <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  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</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>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</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>assume </b><a NAME="E1:63_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:63_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>6</sub></font>,0 <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/><a NAME="E3:63_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T46">CARD_1:46</a></i>;<br/><a NAME="E4:63_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</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="polynom7.html#T1">POLYNOM7:1</a></i>;<br/><a NAME="E5:63_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</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/><b>hence </b><a NAME="E6:63_1"/>
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</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="ref" href="rewrite1.html#T13">REWRITE1:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:63"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:63_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</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>assume </b><a NAME="E1:63_2_1"/><i><font color="Green">E57</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R2">=</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="E2:63_2_1_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</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="E3:63_2_1_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:63_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:63_2_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</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>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="termord.html#K6">Red</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E4:63_2_1"/>
<font color="Maroon">c<sub>8</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="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E5:63_2_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>7</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="E6:63_2_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E7:63_2_1"/><b>then </b>
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="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:63_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E9:63_2_1"/>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="termord.html#T36">TERMORD:36</a></i>;<br/><b>then </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="groeb_3.html#E8:63_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T50">CARD_1:50</a>, <a class="txt" href="groeb_3.html#E1:63_2_1"><i><font color="Green">E57</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 0

;<br/><a NAME="E11:63_2_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</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_3.html#E1:63_2"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E12:63_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</sub></font> <a href="polyred.html#R5">reduces_to</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="groeb_3.html#T47" target="_self">Th47</a></i>;<br/><a NAME="E13:63_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></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">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/><a NAME="E14:63_2_1"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>7</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T16">REWRITE1:16</a></i>;<br/><b>hence </b><a NAME="E15:63_2_1"/>
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>7</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</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="ref" href="rewrite1.html#T17">REWRITE1:17</a>, <a class="txt" href="groeb_3.html#E11:63_2_1"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:63_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:63"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="groeb_3.html#E2:63"><i><font color="Green">E54</font></i></a>, <a class="txt" href="groeb_3.html#E3:63"><i><font color="Green">E55</font></i></a>);<br/></i>
<b>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="E6:63"/><i><font color="Green">E57</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>
<b>thus </b><a NAME="E7:63"/>
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</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_3.html#E4:63"><i><font color="Green">E56</font></i></a>, <a class="txt" href="groeb_3.html#E6:63"><i><font color="Green">E57</font></i></a></i>;<br/>


</div>
