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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>assume </b><a NAME="E1:162"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="ideal_1.html#V9" target="_self">PID</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#D27" target="_self">IDEAL_1:def 27</a><br/>

<a NAME="E2:162"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="ideal_1.html#V8" target="_self">principal</a>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E1:162"><i><font color="Green">E67</font></i></a>, <a class="ref" href="ideal_1.html#D29" target="_self">Def29</a></i>;<br/>
<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="E4:162"/><i><font color="Green">E68</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"><font color="Maroon">c<sub>3</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#D28" target="_self">Def28</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D26" target="_self">IDEAL_1:def 26</a><br/>

<b>thus </b><a NAME="E5:162"/>
<font color="Maroon">c<sub>4</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:162"><i><font color="Green">E68</font></i></a></i>;<br/>


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