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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <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">F</font> being  <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">F</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Olive" title="b1">F</font></span>)</span> &amp; ( for <font color="Olive" title="b2">G</font> being  <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">F</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b2">G</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a>  holds <br/><font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b2">G</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span> &amp; ( for <font color="Olive" title="b1">G</font> being  <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b1">G</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a>  holds <br/><font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">G</font> ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">B</font> =  <a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">H</font> = <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:87"/>
<font color="Maroon" title="c2">F</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span>
  <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">G</font> being  <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b1">G</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a>  holds <br/><font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">G</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_1"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:87_1"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T10" target="_self" title="ARMSTRNG:th.10">Th10</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">c</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="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font> implies <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:87_1_1"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">and </span><br/><a NAME="E2:87_1_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E3:87_1_1"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
not <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">c</font> = <font color="Maroon" title="c8">c</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a></span>;<br/><a NAME="E5:87_1_1"/>
 ex <font color="Olive" title="b1">c9</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font> &amp; ( for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c9</font> holds <br/><font color="Olive" title="b3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c9</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:87_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E3:87_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E2:87_1_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E3:87_1_1"><span class="lab"><font color="Green" title="E53">A5</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>
<span class="kw">hence </span><a NAME="E5:87_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_1"><span class="lab"><font color="Green" title="E50">A2</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>

<span class="kw">let </span><font color="Maroon" title="c5">G</font> be   <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c5">G</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a>  implies <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c5">G</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:87"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E3:87"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<font color="Maroon" title="c5">G</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c5">G</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">B9</font> =  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">z</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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span> or <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">GG</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">f</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b3">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Olive" title="b1">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">c</font> </span>}</span> ;<br/>
<a NAME="E4:87"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">f</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b3">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Olive" title="b1">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">c</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font></span>)</span>
 
;<br/>
<a NAME="E5:87"/>
(  <a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a>  &amp;  <a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#NV2" title="ARMSTRNG:NV.2">(B2)</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T38" target="_self" title="ARMSTRNG:th.38">Th38</a></span>;<br/>
<a NAME="E6:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A9</font></span>: 
 <a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p2">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T37" target="_self" title="ARMSTRNG:th.37">Th37</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:87"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E9:87"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:87"/><span class="lab"><font color="Green" title="E54">A11</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c9">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c10">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font>
 ;<br/>
<a NAME="E11:87"/>
(  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font> is  <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a>  &amp;  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font> is  <a href="armstrng.html#NV2" title="ARMSTRNG:NV.2">(B2)</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="armstrng.html#T34" target="_self" title="ARMSTRNG:th.34">Th34</a></span>;<br/>
<a NAME="E12:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A12</font></span>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">f</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b3">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Olive" title="b1">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">c</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="armstrng.html#T37" target="_self" title="ARMSTRNG:th.37">Th37</a></span>;<br/>
<a NAME="E13:87"/>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p2">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">d</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="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font> or <font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p2">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:87_2"/><span class="lab"><font color="Green" title="E56">A13</font></span>: 
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E2:87_2"/><span class="lab"><font color="Green" title="E57">A14</font></span>: 
not <font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p2">(<span class="default"><a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">d</font> = <font color="Maroon" title="c11">d</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2"><span class="lab"><font color="Green" title="E56">A13</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:87_2"/><span class="lab"><font color="Green" title="E58">A15</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">and </span><br/><a NAME="E6:87_2"/><span class="lab"><font color="Green" title="E59">A16</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">d</font>
 <span class="kw">and </span><br/><a NAME="E7:87_2"/><span class="lab"><font color="Green" title="E60">A17</font></span>: 
not <font color="Maroon" title="c14">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E2:87_2"><span class="lab"><font color="Green" title="E57">A14</font></span></a></span>;<br/>
<a NAME="E8:87_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E5:87_2"><span class="lab"><font color="Green" title="E58">A15</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">e</font>, <font color="Maroon" title="c16">f</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E10:87_2"/><span class="lab"><font color="Green" title="E61">A18</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c15">e</font>,<font color="Maroon" title="c16">f</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E11:87_2"/><span class="lab"><font color="Green" title="E62">A19</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c15">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c16">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:87"><span class="lab"><font color="Green" title="E55">A12</font></span></a></span>;<br/>
<a NAME="E12:87_2"/><span class="lab"><font color="Green" title="E63">A20</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87_2"><span class="lab"><font color="Green" title="E61">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E13:87_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87_2"><span class="lab"><font color="Green" title="E61">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:87_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E6:87_2"><span class="lab"><font color="Green" title="E59">A16</font></span></a>, <a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E60">A17</font></span></a>, <a class="txt" href="#E11:87_2"><span class="lab"><font color="Green" title="E62">A19</font></span></a>, <a class="txt" href="#E12:87_2"><span class="lab"><font color="Green" title="E63">A20</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>
<a NAME="E14:87"/><span class="kw">then </span>
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c9">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c10">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E52">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:87"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="txt" href="#E12:87"><span class="lab"><font color="Green" title="E55">A12</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>
