<?xml version="1.0"?>
<div class="add">

<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</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="b2">R1</font>, <font color="Olive" title="b3">R2</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="b2">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">R2</font></span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Olive" title="b1">X</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R1</font>, <font color="Olive" title="b2">R2</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">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">R2</font></span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">R1</font>, <font color="Maroon" title="c5">R2</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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span></span><br/>









<span class="kw">thus </span><a NAME="E1:40"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<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="firebrick">:: 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="firebrick">::  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="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font> or <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="c4">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1"/><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="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c2">B</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c6">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<a NAME="E4:40_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c4">R1</font>,<font color="Olive" title="b1">x</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c4">R1</font>,<font color="Maroon" title="c9">x</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c4">R1</font>,<font color="Maroon" title="c9">x</font>)</span><br/>

<a NAME="E2:40_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span>,<font color="Maroon" title="c9">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="relset_2.html#T24" target="_self" title="RELSET_2:th.24">Th24</a></span>;<br/>
<a NAME="E3:40_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c4">R1</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c5">R2</font>,<font color="Maroon" title="c9">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T11" target="_self" title="RELSET_2:th.11">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:40_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c4">R1</font>,<font color="Maroon" title="c9">x</font>)
 <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T25" target="_self" title="RELSET_2:th.25">Th25</a></span>;<br/>
<a NAME="E6:40_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c5">R2</font>,<font color="Olive" title="b1">x</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c5">R2</font>,<font color="Maroon" title="c9">x</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c5">R2</font>,<font color="Maroon" title="c9">x</font>)</span><br/>

<a NAME="E2:40_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span>,<font color="Maroon" title="c9">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="relset_2.html#T24" target="_self" title="RELSET_2:th.24">Th24</a></span>;<br/>
<a NAME="E3:40_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c4">R1</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c5">R2</font>,<font color="Maroon" title="c9">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T11" target="_self" title="RELSET_2:th.11">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:40_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c5">R2</font>,<font color="Maroon" title="c9">x</font>)
 <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:40_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T25" target="_self" title="RELSET_2:th.25">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:40_1"/>
<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="c4">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<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="firebrick">:: 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="firebrick">::  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="c4">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> or <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="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:40"/><span class="lab"><font color="Green" title="E19">A3</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="c4">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span><br/>

<a NAME="E3:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">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="c4">R1</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font>
 
<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/>
<a NAME="E4:40"/><span class="lab"><font color="Green" title="E21">A5</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">R2</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c2">B</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c6">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>;<br/>
<a NAME="E7:40"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span>,<font color="Olive" title="b1">x</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span>,<font color="Maroon" title="c9">x</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_2"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span>,<font color="Maroon" title="c9">x</font>)</span><br/>

<a NAME="E2:40_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c4">R1</font>,<font color="Maroon" title="c9">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="relset_2.html#T25" target="_self" title="RELSET_2:th.25">Th25</a></span>;<br/>
<a NAME="E3:40_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c5">R2</font>,<font color="Maroon" title="c9">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="relset_2.html#T25" target="_self" title="RELSET_2:th.25">Th25</a></span>;<br/>
<a NAME="E4:40_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c4">R1</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<font color="Maroon" title="c5">R2</font>,<font color="Maroon" title="c9">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <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="E5:40_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span>,<font color="Maroon" title="c9">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T11" target="_self" title="RELSET_2:th.11">Th11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:40"/>
<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="c4">R1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">R2</font></span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T25" target="_self" title="RELSET_2:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
