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<b>let </b><font color="Maroon">F</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">I</font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">F</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D27">IDEAL_1:def 27</a><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1"/>
( <font color="Maroon">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">F</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> )
 </span><b>by </b><i><a class="ref" href="ideal_1.html#T22">IDEAL_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>suppose </b></a><a NAME="E1:43_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">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">F</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">s0F</font> = <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">F</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F</font> ;<br/><a NAME="E3:43_1_1"/>
<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">F</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">s0F</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T47">IDEAL_1:47</a></i>;<br/><b>hence </b><a NAME="E4:43_1_1"/>
<font color="Maroon">I</font> is <a href="ideal_1.html#V6">finitely_generated</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2"><b>suppose </b></a><a NAME="E1:43_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">s1F</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F</font> ;<br/><a NAME="E3:43_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1F</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T51">IDEAL_1:51</a></i>;<br/><b>hence </b><a NAME="E4:43_1_2"/>
<font color="Maroon">I</font> is <a href="ideal_1.html#V6">finitely_generated</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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