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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Olive" title="b1">X1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Olive" title="b2">X2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">X2</font></span>)</span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">y</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">X2</font></span>)</span> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span> )</span><br/>











<span class="kw">assume </span><a NAME="E1:41"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">X2</font></span>)</span>
 ; <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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><br/>

<a NAME="E2:41"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<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#K9" title="SUBSET_1:func.9">/\</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="subset_1.html#K9" title="SUBSET_1:func.9">/\</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T4" target="_self" title="RELSET_2:th.4">Th4</a></span>;<br/>
<a NAME="E3:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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#K9" title="SUBSET_1:func.9">/\</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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 class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T154" title="RELAT_1:th.154">RELAT_1:154</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:41_1"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X1</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">X2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X1</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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><br/><div class="add"><a NAME="E2:41_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:41_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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#K9" title="SUBSET_1:func.9">/\</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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#K9" title="SUBSET_1:func.9">/\</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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_1_1_1_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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#K9" title="SUBSET_1:func.9">/\</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:41_1_1_1_1_1"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<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#K9" title="SUBSET_1:func.9">/\</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="subset_1.html#K9" title="SUBSET_1:func.9">/\</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T4" target="_self" title="RELSET_2:th.4">Th4</a></span>;<br/><a NAME="E3:41_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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#K9" title="SUBSET_1:func.9">/\</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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 class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T154" title="RELAT_1:th.154">RELAT_1:154</a></span>;<br/><a NAME="E4:41_1_1_1_1_1"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E2:41_1_1_1_1_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="relat_1.html#T154" title="RELAT_1:th.154">RELAT_1:154</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></span>;<br/><a NAME="E5:41_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E3:41_1_1_1_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a></span>;<br/><a NAME="E6:41_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></span>;<br/><a NAME="E7:41_1_1_1_1_1"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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> holds <br/><font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">Y</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="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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>
 ; <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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font></span><br/>

<a NAME="E2:41_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:41_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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><a NAME="E8:41_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/><a NAME="E9:41_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></span>;<br/><a NAME="E10:41_1_1_1_1_1"/>
 <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#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><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> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T154" title="RELAT_1:th.154">RELAT_1:154</a>, <a class="ref" href="setfam_1.html#T59" title="SETFAM_1:th.59">SETFAM_1:59</a></span>;<br/><span class="kw">hence </span><a NAME="E11:41_1_1_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E9:41_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A10</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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#K9" title="SUBSET_1:func.9">/\</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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:41_1_1_1_1_2"/><span class="kw">then </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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:41_1_1_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1"><span class="lab"><font color="Green" title="E22">A4</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:41_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_2"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">X2</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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><br/><div class="add"><a NAME="E2:41_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:41_1_2_1_1"/>
( <span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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#K9" title="SUBSET_1:func.9">/\</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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#K9" title="SUBSET_1:func.9">/\</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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_1_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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#K9" title="SUBSET_1:func.9">/\</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:41_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="relat_1.html#T154" title="RELAT_1:th.154">RELAT_1:154</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></span>;<br/><a NAME="E3:41_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E1:41_1_2_1_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a></span>;<br/><a NAME="E4:41_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A14</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></span>;<br/><a NAME="E5:41_1_2_1_1_1"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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> holds <br/><font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">Y</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="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1_2_1_1_1_1"/>
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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>
 ; <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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font></span><br/>

<a NAME="E2:41_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:41_1_2_1_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_2_1_1_1"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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><a NAME="E6:41_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_2_1_1_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/><a NAME="E7:41_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A15</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_2_1_1_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></span>;<br/><a NAME="E8:41_1_2_1_1_1"/><span class="lab"><font color="Green" title="E27">A16</font></span>: 
<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#K9" title="SUBSET_1:func.9">/\</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="subset_1.html#K9" title="SUBSET_1:func.9">/\</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T4" target="_self" title="RELSET_2:th.4">Th4</a></span>;<br/><a NAME="E9:41_1_2_1_1_1"/>
 <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#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><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> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T154" title="RELAT_1:th.154">RELAT_1:154</a>, <a class="ref" href="setfam_1.html#T59" title="SETFAM_1:th.59">SETFAM_1:59</a></span>;<br/><span class="kw">hence </span><a NAME="E10:41_1_2_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:41_1_2_1_1_1"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="txt" href="#E8:41_1_2_1_1_1"><span class="lab"><font color="Green" title="E27">A16</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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#K9" title="SUBSET_1:func.9">/\</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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:41_1_2_1_1_2"/><span class="kw">then </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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:41_1_2_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="relset_2.html#K4" title="RELSET_2:func.4">.:</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2"><span class="lab"><font color="Green" title="E22">A11</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:41_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span>)</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></div><span class="kw">end;</span></div>

</div>
