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<b>let </b><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c1">Y</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">Y'</font> = <font color="Maroon" title="c1">Y</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="prob_1.html#E1:33"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/>
<a NAME="E3:33"/>
<font color="Maroon" title="c1">Y</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">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" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a><br/>

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E23">A2</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">A'</font> = <font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c7">B'</font> = <font color="Maroon" title="c5">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="prob_1.html#E1:33"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">A1</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E4:33_1"/><i><font color="Green" title="E24">A3</font></i>: 
( <font color="Maroon" title="c8">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A'</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 holds <br/><font color="Maroon" title="c8">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B'</font> ) )
 <b>by </b><i><a class="ref" href="prob_1.html#T22" target="_self" title="PROB_1:th.22">Th22</a></i>;<br/>
<a NAME="E5:33_1"/><i><font color="Green" title="E25">A4</font></i>: 
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="txt" href="prob_1.html#E4:33_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="prob_1.html#T30" target="_self" title="PROB_1:th.30">Th30</a></i>;<br/>
<a NAME="E6:33_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c8">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="txt" href="prob_1.html#E1:33_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="prob_1.html#E4:33_1"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:33_1"/>
<font color="Maroon" title="c4">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="prob_1.html#E1:33"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="prob_1.html#E5:33_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="prob_1.html#D8" target="_self" title="PROB_1:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33"/>
<font color="Maroon" title="c1">Y</font> is   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="prob_1.html#E1:33"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/>


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