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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="E1:30"/><i><font color="Green" title="E19">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/>

<a NAME="E2:30"/>
<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="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">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="E1:30_1"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c3">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="c4">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="c5">A'</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">B'</font> = <font color="Maroon" title="c4">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#E1:30"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">S</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c6">B'</font> as   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">A1</font> = <font color="Maroon" title="c7">S</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c5">A'</font> as   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="ref" href="funct_7.html#T117" title="FUNCT_7:th.117">FUNCT_7:117</a></i>;<br/>
<a NAME="E5:30_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E6:30_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c6">B'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E7:30_1"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c8">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A'</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<a NAME="E8:30_1"/><i><font color="Green" title="E22">A4</font></i>: 
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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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="c6">B'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">n</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>hence </b><font color="Maroon" title="c8">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font> = 
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c6">B'</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
<b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon" title="c6">B'</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E9:30_1"/><i><font color="Green" title="E23">A5</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="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="prob_1.html#E7:30_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="prob_1.html#E8:30_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="prob_1.html#T30" target="_self" title="PROB_1:th.30">Th30</a></i>;<br/>
<a NAME="E10:30_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#E1:30_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="prob_1.html#E7:30_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="prob_1.html#E8:30_1"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:30_1"/>
<font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">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#E1:30"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="prob_1.html#E9:30_1"><i><font color="Green" title="E23">A5</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="E3:30"/>
<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#E1:30"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>


</div>
