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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b3">R</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <span class="p1">(<span class="default"><a href="relset_2.html#K3" title="RELSET_2:func.3">.:</a> <font color="Olive" title="b3">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Olive" title="b1">X1</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><font color="Olive" title="b3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Olive" title="b2">X2</font></span><br/><b>let </b><font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <span class="p1">(<span class="default"><a href="relset_2.html#K3" title="RELSET_2:func.3">.:</a> <font color="Olive" title="b1">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c3">X1</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><font color="Olive" title="b1">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">X2</font></span><br/><b>let </b><font color="Maroon" title="c5">R</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="relset_2.html#K3" title="RELSET_2:func.3">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c3">X1</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">X2</font> )</span><br/>









<a NAME="E1:34"/><i><font color="Green" title="E17">A1</font></i>: 
<span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c3">X1</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c4">X2</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span>
 
<b>by </b><i><a class="ref" href="relset_2.html#T3" target="_self" title="RELSET_2:th.3">Th3</a></i>;<br/>
<b>assume </b><a NAME="E2:34"/><i><font color="Green" title="E18">A2</font></i>: 
<span class="p1">(<span class="default"><a href="relset_2.html#K3" title="RELSET_2:func.3">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c3">X1</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">X2</font></span><br/>

<font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span> = 
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_2.html#K3" title="RELSET_2:func.3">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c3">X1</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_2.html#K3" title="RELSET_2:func.3">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c4">X2</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="relset_2.html#E1:34"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></i>
<br/>.= 
<font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">X2</font>
<b>by </b><i><a class="txt" href="relset_2.html#E2:34"><i><font color="Green" title="E18">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:34"/>
<font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">X2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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