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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> )</span><br/>



<span class="kw">assume </span><a NAME="E1:47"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:47"/>
<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="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or  not <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> or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">A9</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">B9</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">A1</font> = <font color="Maroon" title="c5">A9</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c6">B9</font>;<br/>
<a NAME="E3:47_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A9</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c6">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">B9</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T128" title="FUNCT_7:th.128">FUNCT_7:128</a></span>;<br/>
<a NAME="E4:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A9</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c6">B9</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></span>;<br/>
<a NAME="E5:47_1"/>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A9</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c6">B9</font></span>)</span> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T30" target="_self" title="PROB_1:th.30">Th30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:47_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:47_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="prob_1.html#D8" target="_self" title="PROB_1:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:47"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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