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

<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">B</font> be  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/><b>let </b><font color="Maroon" title="c3">R</font> be    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c5">G</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c2">B</font>;<br/>









<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Olive" title="b1">Y</font></span>)</span> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font> : <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font> <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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:39_1_1"/><b>then </b><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="relset_2.html#T29" target="_self" title="RELSET_2:th.29">Th29</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1"/>
( <font color="Maroon" title="c5">G</font> <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 <font color="Maroon" title="c5">G</font> <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="39_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_1_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">Y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:39_1_1_1_1_1"/><i><font color="Green" title="E22">A5</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 <b>by </b><i><a class="txt" href="relset_2.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<a NAME="E4:39_1_1_1_1_1"/>
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="relset_2.html#E3:39_1_1_1_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="relset_2.html#E1:39_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="orders_1.html#T91" title="ORDERS_1:th.91">ORDERS_1:91</a></i>;<br/>
<a NAME="E5:39_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_2.html#K3" title="RELSET_2:func.3">.:</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c7">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relset_2.html#T23" target="_self" title="RELSET_2:th.23">Th23</a></i>;<br/>
<a NAME="E6:39_1_1_1_1_1"/><b>then </b>
 <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#K3" title="RELSET_2:func.3">.:</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c7">Y</font></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>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:39_1_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="relset_2.html#E3:39_1_1_1_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:39_1_1_1_1"/><b>then </b>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T7" title="SETFAM_1:th.7">SETFAM_1:7</a>, <a class="txt" href="relset_2.html#E1:39_1_1_1_1"><i><font color="Green" title="E21">A4</font></i></a></i>;<br/><a NAME="E4:39_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>;<br/><a NAME="E5:39_1_1_1_1"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E6:39_1_1_1_1"/>
<font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a>, <a class="txt" href="relset_2.html#E1:39_1_1_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="relset_2.html#E2:39_1_1"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_2"/>
<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_1_2"/>
<font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a>, <a class="txt" href="relset_2.html#E2:39_1_1"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">Z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_2"/><i><font color="Green" title="E19">A6</font></i>: 
( <font color="Maroon" title="c6">Z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 <b>by </b><i><a class="ref" href="orders_1.html#T91" title="ORDERS_1:th.91">ORDERS_1:91</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">Z1</font> = <font color="Maroon" title="c6">Z1</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="relset_2.html#E3:39_1_2"><i><font color="Green" title="E19">A6</font></i></a></i>;<br/><a NAME="E5:39_1_2"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_2_1"/>
not <font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:39_1_2_1"/><b>then </b>
not <font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c7">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 
;<br/>
<b>hence </b><a NAME="E3:39_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="relset_2.html#E3:39_1_2"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="relset_2.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E6:39_1_2"/>
<font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></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> <font color="Maroon" title="c5">G</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_1_2_2"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/>

<a NAME="E2:39_1_2_2"/>
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> <font color="Maroon" title="c5">G</font> holds <br/><font color="Maroon" title="c8">a</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="39_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">Z2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_2_2_1"/>
<font color="Maroon" title="c9">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">Z3</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:39_1_2_2_1"/><i><font color="Green" title="E22">A9</font></i>: 
( <font color="Maroon" title="c9">Z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c10">Z3</font> &amp; <font color="Maroon" title="c10">Z3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 <b>by </b><i><a class="txt" href="relset_2.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c8">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">B</font> <b>by </b><i><a class="txt" href="relset_2.html#E1:39_1_2_2"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/>
<a NAME="E5:39_1_2_2_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="c10">Z3</font> holds <br/><font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">R</font>,<font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_2_2_1_1"/><i><font color="Green" title="E23">A10</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z3</font>
 ;<br/>

<a NAME="E2:39_1_2_2_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="relset_2.html#E1:39_1_2_2_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="relset_2.html#E3:39_1_2_2_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_2_2_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="ref" href="relset_2.html#T24" target="_self" title="RELSET_2:th.24">Th24</a>, <a class="txt" href="relset_2.html#E1:39_1_2_2"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:39_1_2_2_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z2</font>
 <b>by </b><i><a class="ref" href="relset_2.html#T25" target="_self" title="RELSET_2:th.25">Th25</a>, <a class="txt" href="relset_2.html#E3:39_1_2_2_1"><i><font color="Green" title="E22">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:39_1_2_2"/><b>then </b>
<font color="Maroon" title="c8">a</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> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="relset_2.html#E5:39_1_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_2_2"/>
<font color="Maroon" title="c8">a</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> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="relset_2.html#E5:39_1_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E7:39_1_2"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c8">a</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> <font color="Maroon" title="c5">G</font>
 ;<br/><a NAME="E8:39_1_2"/><b>then </b><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c8">a</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> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="relset_2.html#E5:39_1_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">B</font> <b>by </b><i><a class="txt" href="relset_2.html#E7:39_1_2"><i><font color="Green" title="E21">A11</font></i></a></i>;<br/><a NAME="E10:39_1_2"/>
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>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font> holds <br/><font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">R</font>,<font color="Olive" title="b1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_2_3"/>
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_2_3"/><i><font color="Green" title="E23">A13</font></i>: 
( <font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Z</font> &amp; <font color="Maroon" title="c11">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">Z</font> = <font color="Maroon" title="c11">Z</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="relset_2.html#E3:39_1_2_3"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">C</font> = <font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c12">Z</font>;<br/>
<a NAME="E5:39_1_2_3"/>
<font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c12">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="relset_2.html#E3:39_1_2_3"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="relset_2.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<a NAME="E6:39_1_2_3"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c12">Z</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a>, <a class="txt" href="relset_2.html#E8:39_1_2"><i><font color="Green" title="E22">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:39_1_2_3"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c10">X</font>
 <b>by </b><i><a class="ref" href="relset_2.html#T24" target="_self" title="RELSET_2:th.24">Th24</a>, <a class="txt" href="relset_2.html#E3:39_1_2_3"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:39_1_2"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="relset_2.html#T25" target="_self" title="RELSET_2:th.25">Th25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
