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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="prob_1.html#E1:33"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E3:33"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="finsub_1.html#V2">cap-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <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="finsub_1.html#D2">FINSUB_1:def 2</a><br/>

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="prob_1.html#E1:33"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:33_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="prob_1.html#K1" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="ref" href="prob_1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E5:33_1"/><i><font color="Green">E25</font></i>: 
 <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prob_1.html#E4:33_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="prob_1.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E6:33_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>8</sub></font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="prob_1.html#E1:33_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="prob_1.html#E4:33_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:33_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="prob_1.html#E1:33"><i><font color="Green">E22</font></i></a>, <a class="txt" href="prob_1.html#E5:33_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="prob_1.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="prob_1.html#E1:33"><i><font color="Green">E22</font></i></a></i>;<br/>


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