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<b>thus </b><a NAME="E1:30_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="nagata_1.html#V3" target="_self">sigma_locally_finite</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:30_1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="nagata_1.html#V2" target="_self">sigma_discrete</a>
 <b>by </b><i><a class="txt" href="nagata_1.html#E18"><i><font color="Green">Lemma12</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:30_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="nagata_1.html#V1" target="_self">discrete</a>
 <b>by </b><i><a class="txt" href="nagata_1.html#E2:30_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="nagata_1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E2:30_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pcomps_1.html#V1">locally_finite</a>
 <b>by </b><i><a class="ref" href="nagata_1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="nagata_1.html#V3" target="_self">sigma_locally_finite</a>
 <b>by </b><i><a class="ref" href="nagata_1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>


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