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<b>let </b><font color="Maroon">R</font> be  non <a href="vectsp_1.html#V10">degenerated</a> <a href="vectsp_2.html#NM1">comRing</a>;<br/>

<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">R</font> is   <a href="vectsp_1.html#NM2">Field</a>
 ;<br/><b>thus </b><a NAME="E2:45_1"/>
for <font color="Olive">I</font> being  <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>  holds <br/>( <font color="Olive">I</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">R</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Olive">I</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">I</font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">I</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">R</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="vectsp_1.html#NM2">Field</a> <b>by </b><i/>;<br/>
<a NAME="E3:45_1_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  holds <br/>(  not <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> or  not <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/>

<div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> ;<br/><a NAME="E3:45_1_1_1_1"/>
<font color="Maroon">u'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:45_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</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">R</font></span>)</span></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/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</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">R</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/><a NAME="E3:45_1_1_1_2"/>
<font color="Maroon">u'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:45_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:45_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:45_1_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E7:45_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#D20">VECTSP_1:def 20</a></i>;<br/>
<a NAME="E8:45_1_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:45_1_1"/><b>then </b>
not <font color="Maroon">I</font> is <a href="ideal_1.html#V5" target="_self">proper</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:45_1_1"/>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_2"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">I</font> being  <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>  holds <br/>( <font color="Olive">I</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">R</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Olive">I</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:45_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/><b>reconsider </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="group_1.html#K10">*</a> <font color="Olive">r</font></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span>  as   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><a NAME="E4:45_2_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/><a NAME="E5:45_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E6:45_2_1"/><b>then </b>
<font color="Maroon">M</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">R</font></span>)</span></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="E7:45_2_1"/><b>then </b>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><a NAME="E8:45_2_1"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E10:45_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 ;<br/><b>thus </b><a NAME="E11:45_2_1"/>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:45_2"/>
<font color="Maroon">R</font> is   <a href="vectsp_1.html#NM2">Field</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D20">VECTSP_1:def 20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:45"/>
( <font color="Maroon">R</font> is   <a href="vectsp_1.html#NM2">Field</a> iff for <font color="Olive">I</font> being  <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>  holds <br/>( <font color="Olive">I</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">R</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Olive">I</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> ) )
 <b>by </b><i/>;<br/>


</div>
