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<div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">B</font> is <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a> &amp; <font color="Olive" title="b1">B</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">(B2)</a> holds <br/>( <font color="Olive" title="b1">B</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> <font color="Olive" title="b1">B</font></span>)</span> &amp; ( for <font color="Olive" title="b2">G</font> being  <a href="armstrng.html#NM5" title="ARMSTRNG:NM.5">Full-family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b2">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Olive" title="b1">B</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">B</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">B</font> is <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a> &amp; <font color="Maroon" title="c2">B</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">(B2)</a> implies ( <font color="Maroon" title="c2">B</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> <font color="Maroon" title="c2">B</font></span>)</span> &amp; ( for <font color="Olive" title="b1">G</font> being  <a href="armstrng.html#NM5" title="ARMSTRNG:NM.5">Full-family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Olive" title="b1">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> ) ) )</span><br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E46">A1</font></i>: 
( <font color="Maroon" title="c2">B</font> is <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a> &amp; <font color="Maroon" title="c2">B</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">(B2)</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">B</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> <font color="Maroon" title="c2">B</font></span>)</span> &amp; ( for <font color="Olive" title="b1">G</font> being  <a href="armstrng.html#NM5" title="ARMSTRNG:NM.5">Full-family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Olive" title="b1">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> ) )</span><br/>

<b>set </b><font color="Maroon" title="c3">F</font> = <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>;<br/>
<b>set </b><font color="Maroon" title="c4">ss</font> =  <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> <font color="Maroon" title="c2">B</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">F'</font> = <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> as   <a href="armstrng.html#NM5" title="ARMSTRNG:NM.5">Full-family</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="armstrng.html#T35" target="_self" title="ARMSTRNG:th.35">Th35</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">M</font> =  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c5">F'</font>;<br/>
<a NAME="E3:81"/><i><font color="Green" title="E47">A2</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="armstrng.html#D4" target="_self" title="ARMSTRNG:def.4">Def4</a></i>;<br/>
<b>reconsider </b><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>  <b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="armstrng.html#D4" target="_self" title="ARMSTRNG:def.4">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c8">x</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> <font color="Maroon" title="c2">B</font></span>)</span> ) &amp; ( <font color="Maroon" title="c8">x</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> <font color="Maroon" title="c2">B</font></span>)</span> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) )</span><br/><a NAME="E1:81_1"/>
<font color="Maroon" title="c2">B</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> <font color="Maroon" title="c2">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="armstrng.html#T36" target="_self" title="ARMSTRNG:th.36">Th36</a></i>;<br/><b>hence </b><a NAME="E2:81_1"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c8">x</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> <font color="Maroon" title="c2">B</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x</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> <font color="Maroon" title="c2">B</font></span>)</span> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )</span><br/><b>assume </b><a NAME="E3:81_1"/>
<font color="Maroon" title="c8">x</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> <font color="Maroon" title="c2">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font></span><br/><b>then consider </b><font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c10">a</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:81_1"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 <b>and </b><br/><a NAME="E6:81_1"/><i><font color="Green" title="E49">A4</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="armstrng.html#T33" target="_self" title="ARMSTRNG:th.33">Th33</a></i>;<br/><a NAME="E7:81_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c9">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c5">F'</font>
 <b>by </b><i><a class="txt" href="#E6:81_1"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/><a NAME="E8:81_1"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c9">b</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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">aa</font>, <font color="Maroon" title="c12">bb</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E10:81_1"/><i><font color="Green" title="E50">A5</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c9">b</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="c11">aa</font>,<font color="Maroon" title="c12">bb</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E11:81_1"/><i><font color="Green" title="E51">A6</font></i>: 
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> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c11">aa</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="c12">bb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font>
 ;<br/><a NAME="E12:81_1"/><i><font color="Green" title="E52">A7</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">aa</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">bb</font> )
 <b>by </b><i><a class="txt" href="#E10:81_1"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>set </b><font color="Maroon" title="c13">S</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b'</font> where <font color="Olive" title="b1">b'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> : <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b'</font> </span>}</span> ;<br/><a NAME="E13:81_1"/><i><font color="Green" title="E53">A9</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">b'</font> where <font color="Olive" title="b1">b'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> : <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b'</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b'</font> where <font color="Olive" title="b1">b'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> : <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b'</font> </span>}</span>  or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:81_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b'</font> where <font color="Olive" title="b1">b'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> : <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b'</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span><br/>

<b>then consider </b><font color="Maroon" title="c15">b'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font><b> such that </b><br/><a NAME="E3:81_1_1"/><i><font color="Green" title="E54">A10</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b'</font> &amp; <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">b'</font> )
 ;<br/>
<a NAME="E4:81_1_1"/>
( <font color="Maroon" title="c15">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B'</font> &amp; <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> )
 
;<br/>
<b>hence </b><a NAME="E5:81_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E3:81_1_1"><i><font color="Green" title="E54">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E14:81_1"/><i><font color="Green" title="E54">A12</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b'</font> where <font color="Olive" title="b1">b'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> : <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b'</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E3:81"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><a NAME="E15:81_1"/><i><font color="Green" title="E55">A13</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">b'</font> where <font color="Olive" title="b1">b'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> : <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b'</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b'</font> where <font color="Olive" title="b1">b'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> : <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b'</font> </span>}</span>  or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<b>assume </b><a NAME="E1:81_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b'</font> where <font color="Olive" title="b1">b'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> : <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b'</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font></span><br/>

<b>then consider </b><font color="Maroon" title="c15">b'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font><b> such that </b><br/><a NAME="E3:81_1_2"/><i><font color="Green" title="E56">A14</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b'</font> &amp; <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">b'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:81_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="#E3:81_1_2"><i><font color="Green" title="E56">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c14">S</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b'</font> where <font color="Olive" title="b1">b'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> : <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b'</font> </span>}</span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="#E13:81_1"><i><font color="Green" title="E53">A9</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c15">mS</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c14">S</font>;<br/><b>reconsider </b><font color="Maroon" title="c16">mS</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c14">S</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E18:81_1"/><i><font color="Green" title="E56">A15</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">mS</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">mS</font> )</span><br/>

<b>assume </b><a NAME="E1:81_1_3"/><i><font color="Green" title="E57">A16</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">mS</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> implies <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font> )</span><br/><b>assume </b><a NAME="E1:81_1_3_1"/>
<font color="Maroon" title="c18">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font></span><br/><b>then consider </b><font color="Maroon" title="c19">b'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font><b> such that </b><br/><a NAME="E3:81_1_3_1"/><i><font color="Green" title="E58">A17</font></i>: 
( <font color="Maroon" title="c18">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b'</font> &amp; <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">b'</font> )
 ;<br/><a NAME="E4:81_1_3_1"/>
<font color="Maroon" title="c9">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">b'</font>
 <b>by </b><i><a class="txt" href="#E11:81_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="#E12:81_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="#E3:81_1_3_1"><i><font color="Green" title="E58">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E5:81_1_3_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font>
 <b>by </b><i><a class="txt" href="#E1:81_1_3"><i><font color="Green" title="E57">A16</font></i></a>, <a class="txt" href="#E3:81_1_3_1"><i><font color="Green" title="E58">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:81_1_3"/><b>then </b>
<font color="Maroon" title="c17">x</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="c14">S</font>
 
<b>by </b><i><a class="txt" href="#E14:81_1"><i><font color="Green" title="E54">A12</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="E4:81_1_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">mS</font>
 <b>by </b><i><a class="txt" href="#E14:81_1"><i><font color="Green" title="E54">A12</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_1_4"/><i><font color="Green" title="E57">A18</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">mS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">c</font> implies <font color="Maroon" title="c16">mS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">c</font> )</span><br/><b>assume </b><a NAME="E1:81_1_4_1"/>
( <font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">mS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">c</font></span><br/><a NAME="E2:81_1_4_1"/><b>then </b>
<font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font>
 ;<br/><a NAME="E3:81_1_4_1"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c14">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">c</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/><b>hence </b><a NAME="E4:81_1_4_1"/>
<font color="Maroon" title="c16">mS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">c</font>
 <b>by </b><i><a class="txt" href="#E14:81_1"><i><font color="Green" title="E54">A12</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E3:81_1_4"/><b>then </b><i><font color="Green" title="E58">A19</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c16">mS</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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ;<br/><a NAME="E4:81_1_4"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c16">mS</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c9">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="txt" href="#E18:81_1"><i><font color="Green" title="E56">A15</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E5:81_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:81_1"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="#E1:81_1_4"><i><font color="Green" title="E57">A18</font></i></a>, <a class="txt" href="#E3:81_1_4"><i><font color="Green" title="E58">A19</font></i></a>, <a class="ref" href="armstrng.html#T29" target="_self" title="ARMSTRNG:th.29">Th29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E20:81_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="#E3:81"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="#E5:81_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="#E15:81_1"><i><font color="Green" title="E55">A13</font></i></a>, <a class="ref" href="armstrng.html#T1" target="_self" title="ARMSTRNG:th.1">Th1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:81"/>
<font color="Maroon" title="c2">B</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> <font color="Maroon" title="c2">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="armstrng.html#NM5" title="ARMSTRNG:NM.5">Full-family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Olive" title="b1">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font></span><br/>

<b>let </b><font color="Maroon" title="c8">G</font> be   <a href="armstrng.html#NM5" title="ARMSTRNG:NM.5">Full-family</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c8">G</font> implies <font color="Maroon" title="c8">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<b>assume </b><a NAME="E7:81"/><i><font color="Green" title="E48">A20</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c8">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font></span><br/>

<b>set </b><font color="Maroon" title="c9">MG</font> =  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c8">G</font>;<br/>
<a NAME="E8:81"/><i><font color="Green" title="E49">A21</font></i>: 
(  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c8">G</font> is <a href="armstrng.html#V8" title="ARMSTRNG:attr.8">(M1)</a> &amp;  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c8">G</font> is <a href="armstrng.html#V10" title="ARMSTRNG:attr.10">(M3)</a> )
 
<b>by </b><i><a class="ref" href="armstrng.html#T30" target="_self" title="ARMSTRNG:th.30">Th30</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>now </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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> implies <font color="Maroon" title="c10">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> <font color="Maroon" title="c2">B</font> ) &amp; ( <font color="Maroon" title="c10">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> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">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> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:81_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">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> <font color="Maroon" title="c2">B</font></span><br/><b>then reconsider </b><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c10">x</font> as    <a href="armstrng.html#M1" title="ARMSTRNG:mode.1">Element</a> of <font color="Maroon" title="c8">G</font> ;<br/><b>reconsider </b><font color="Maroon" title="c12">x'</font> = <font color="Maroon" title="c11">x1</font> as   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font> ;<br/><b>consider </b><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:81_2_1"/><i><font color="Green" title="E50">A22</font></i>: 
<font color="Maroon" title="c12">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="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T9" target="_self" title="ARMSTRNG:th.9">Th9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">b'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B'</font> &amp; <font color="Maroon" title="c13">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">b'</font> implies <font color="Maroon" title="c14">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">b'</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:81_2_1_1"/><i><font color="Green" title="E51">A23</font></i>: 
<font color="Maroon" title="c15">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B'</font>
 <b>and </b><br/><a NAME="E2:81_2_1_1"/><i><font color="Green" title="E52">A24</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">b'</font></span><br/><b>consider </b><font color="Maroon" title="c16">b1'</font>, <font color="Maroon" title="c17">a'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:81_2_1_1"/><i><font color="Green" title="E53">A25</font></i>: 
<font color="Maroon" title="c15">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b1'</font>
 <b>and </b><br/><a NAME="E5:81_2_1_1"/><i><font color="Green" title="E54">A26</font></i>: 
<font color="Maroon" title="c17">a'</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c16">b1'</font>,<font color="Maroon" title="c8">G</font>
 <b>by </b><i><a class="txt" href="#E7:81"><i><font color="Green" title="E48">A20</font></i></a>, <a class="txt" href="#E1:81_2_1_1"><i><font color="Green" title="E51">A23</font></i></a>, <a class="ref" href="armstrng.html#T33" target="_self" title="ARMSTRNG:th.33">Th33</a></i>;<br/><a NAME="E6:81_2_1_1"/><i><font color="Green" title="E55">A27</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c15">b'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c8">G</font>
 <b>by </b><i><a class="txt" href="#E4:81_2_1_1"><i><font color="Green" title="E53">A25</font></i></a>, <a class="txt" href="#E5:81_2_1_1"><i><font color="Green" title="E54">A26</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">a''</font>, <font color="Maroon" title="c19">b''</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:81_2_1_1"/><i><font color="Green" title="E56">A28</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c18">a''</font>,<font color="Maroon" title="c19">b''</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c8">G</font>
 <b>and </b><br/><a NAME="E9:81_2_1_1"/><i><font color="Green" title="E57">A29</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c18">a''</font>,<font color="Maroon" title="c19">b''</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <font color="Maroon" title="c12">x'</font>
 <b>by </b><i><a class="ref" href="armstrng.html#T28" target="_self" title="ARMSTRNG:th.28">Th28</a></i>;<br/><a NAME="E10:81_2_1_1"/>
<font color="Maroon" title="c18">a''</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">a</font>
 <b>by </b><i><a class="txt" href="#E5:81_2_1"><i><font color="Green" title="E50">A22</font></i></a>, <a class="txt" href="#E9:81_2_1_1"><i><font color="Green" title="E57">A29</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><a NAME="E11:81_2_1_1"/><b>then </b>
<font color="Maroon" title="c18">a''</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">b'</font>
 <b>by </b><i><a class="txt" href="#E2:81_2_1_1"><i><font color="Green" title="E52">A24</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:81_2_1_1"/><b>then </b><i><font color="Green" title="E58">A30</font></i>: 
<font color="Maroon" title="c19">b''</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">b1'</font>
 <b>by </b><i><a class="txt" href="#E8:81"><i><font color="Green" title="E49">A21</font></i></a>, <a class="txt" href="#E4:81_2_1_1"><i><font color="Green" title="E53">A25</font></i></a>, <a class="txt" href="#E6:81_2_1_1"><i><font color="Green" title="E55">A27</font></i></a>, <a class="txt" href="#E8:81_2_1_1"><i><font color="Green" title="E56">A28</font></i></a>, <a class="ref" href="armstrng.html#D21" target="_self" title="ARMSTRNG:def.21">Def21</a></i>;<br/><a NAME="E13:81_2_1_1"/>
<font color="Maroon" title="c14">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">b''</font>
 <b>by </b><i><a class="txt" href="#E5:81_2_1"><i><font color="Green" title="E50">A22</font></i></a>, <a class="txt" href="#E9:81_2_1_1"><i><font color="Green" title="E57">A29</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E14:81_2_1_1"/>
<font color="Maroon" title="c14">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">b'</font>
 <b>by </b><i><a class="txt" href="#E4:81_2_1_1"><i><font color="Green" title="E53">A25</font></i></a>, <a class="txt" href="#E12:81_2_1_1"><i><font color="Green" title="E58">A30</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E7:81_2_1"/>
<font color="Maroon" title="c10">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> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="#E5:81_2_1"><i><font color="Green" title="E50">A22</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:81_2"/>
<font color="Maroon" title="c10">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> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font></span><br/><b>then consider </b><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:81_2"/><i><font color="Green" title="E50">A31</font></i>: 
<font color="Maroon" title="c10">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="c11">a</font>,<font color="Maroon" title="c12">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:81_2"/><i><font color="Green" title="E51">A32</font></i>: 
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> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c11">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="c12">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font>
 ;<br/><b>consider </b><font color="Maroon" title="c13">a'</font>, <font color="Maroon" title="c14">b'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E7:81_2"/><i><font color="Green" title="E52">A33</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c11">a</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E8:81_2"/><i><font color="Green" title="E53">A34</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c8">G</font>
 <b>by </b><i><a class="txt" href="#E8:81"><i><font color="Green" title="E49">A21</font></i></a>, <a class="ref" href="armstrng.html#D19" target="_self" title="ARMSTRNG:def.19">Def19</a></i>;<br/><a NAME="E9:81_2"/><i><font color="Green" title="E54">A35</font></i>: 
( <font color="Maroon" title="c13">a'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">a</font> &amp; <font color="Maroon" title="c11">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">b'</font> )
 <b>by </b><i><a class="txt" href="#E7:81_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><a NAME="E10:81_2"/>
<font color="Maroon" title="c13">a'</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c8">G</font>
 <b>by </b><i><a class="txt" href="#E8:81_2"><i><font color="Green" title="E53">A34</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/><a NAME="E11:81_2"/><b>then </b>
<font color="Maroon" title="c14">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="#E7:81"><i><font color="Green" title="E48">A20</font></i></a></i>;<br/><a NAME="E12:81_2"/><b>then </b>
<font color="Maroon" title="c12">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">b'</font>
 <b>by </b><i><a class="txt" href="#E5:81_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="txt" href="#E9:81_2"><i><font color="Green" title="E54">A35</font></i></a></i>;<br/><a NAME="E13:81_2"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="txt" href="#E9:81_2"><i><font color="Green" title="E54">A35</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E14:81_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font>
 <b>by </b><i><a class="txt" href="#E4:81_2"><i><font color="Green" title="E50">A31</font></i></a>, <a class="txt" href="#E8:81_2"><i><font color="Green" title="E53">A34</font></i></a>, <a class="ref" href="armstrng.html#D13" target="_self" title="ARMSTRNG:def.13">Def13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:81"/>
<font color="Maroon" title="c8">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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