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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</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="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="algstr_1.html#V1">left_zeroed</a> <a href="int_3.html#V1">Euclidian</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D29" target="_self">IDEAL_1:def 29</a><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="161_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:161_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="161_1_1"><b>suppose </b></a><a NAME="E1:161_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>take </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D28" target="_self">IDEAL_1:def 28</a><br/><b>thus </b><a NAME="E2:161_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:161_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="ideal_1.html#T47" target="_self">Th47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="161_1_2"><b>suppose </b></a><a NAME="E1:161_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:161_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="ideal_1.html#T96" target="_self">Th96</a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> ;<br/><a NAME="E4:161_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>3</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="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:161_1_2"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E5:161_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:161_1_2_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="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:161_1_2_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:161_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:161_1_2_1"><i><font color="Green">E68</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ideal_1.html#E4:161_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:161_1_2"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>4</sub></font> &amp; ( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="int_3.html#D9">INT_3:def 9</a></i>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : verum </span>}</span> ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E10:161_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : verum </span>}</span> 
 ;<br/><a NAME="E11:161_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:161_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : verum </span>}</span> 
 ;<br/>

<b>then 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>5</sub></font><b> such that </b><br/><a NAME="E3:161_1_2_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>
<b>thus </b><a NAME="E4:161_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:161_1_2_2"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : verum </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="ideal_1.html#E10:161_1_2"><i><font color="Green">E69</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E13:161_1_2"/>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E15:161_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E16:161_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E18:161_1_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>take </b>
<font color="Maroon">c<sub>13</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D28" target="_self">IDEAL_1:def 28</a><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:161_1_2_3_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:161_1_2_3_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E5:161_1_2_3_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E8:161_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="ideal_1.html#E15:161_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="ideal_1.html#E18:161_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1_2_3_1_1"><b>now </b></a><div class="add"><i><font color="Green">E75</font></i>: <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="ideal_1.html#E4:161_1_2_3_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="algstr_1.html#D5">ALGSTR_1:def 5</a></i>
;<br/><a NAME="E2:161_1_2_3_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E18:161_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="ideal_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E3:161_1_2_3_1_1"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E4:161_1_2_3_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:161_1_2_3_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="ideal_1.html#E1:161_1_2_3_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="ideal_1.html#D1" target="_self">Def1</a></i>;<br/><b>assume </b><a NAME="E5:161_1_2_3_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E6:161_1_2_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:161_1_2_3_1_1"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E7:161_1_2_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>hence </b><a NAME="E8:161_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ideal_1.html#E5:161_1_2_3_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="ideal_1.html#E5:161_1_2_3_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/></div><b>end;</b></div><a NAME="E7:161_1_2_3_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:161_1_2_3_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>;<br/><a NAME="E8:161_1_2_3_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:161_1_2_3_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#D15" target="_self">Def15</a></i>;<br/><b>hence </b><a NAME="E10:161_1_2_3_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:161_1_2_3_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="ideal_1.html#E8:161_1_2_3_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="ideal_1.html#D2" target="_self">Def2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:161_1_2_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 ;<br/><a NAME="E3:161_1_2_3"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E18:161_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E4:161_1_2_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D15" target="_self">Def15</a></i>;<br/><b>hence </b><a NAME="E5:161_1_2_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E2:161_1_2_3"><i><font color="Green">E72</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E21:161_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
