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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">F</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">M</font> is  <a href="armstrng.html#V8" title="ARMSTRNG:attr.8">(M1)</a>  &amp; <font color="Olive" title="b1">M</font> is  <a href="armstrng.html#V9" title="ARMSTRNG:attr.9">(M2)</a>  &amp; <font color="Olive" title="b1">M</font> is  <a href="armstrng.html#V10" title="ARMSTRNG:attr.10">(M3)</a>  &amp; <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="b3">A</font>,<font color="Olive" title="b4">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b5">A9</font>, <font color="Olive" title="b6">B9</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/>( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b5">A9</font>,<font color="Olive" title="b6">B9</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="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b5">A9</font>,<font color="Olive" title="b6">B9</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="Olive" title="b1">M</font> ) </span> } </span>  holds <br/>( <font color="Olive" title="b1">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b2">F</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a>  &amp; (  for <font color="Olive" title="b3">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">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Olive" title="b3">G</font> holds <br/><font color="Olive" title="b3">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">F</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> is  <a href="armstrng.html#V8" title="ARMSTRNG:attr.8">(M1)</a>  &amp; <font color="Maroon" title="c2">M</font> is  <a href="armstrng.html#V9" title="ARMSTRNG:attr.9">(M2)</a>  &amp; <font color="Maroon" title="c2">M</font> is  <a href="armstrng.html#V10" title="ARMSTRNG:attr.10">(M3)</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">A</font>,<font color="Olive" title="b2">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b3">A9</font>, <font color="Olive" title="b4">B9</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/>( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">A9</font>,<font color="Olive" title="b4">B9</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="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">A9</font>,<font color="Olive" title="b4">B9</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">M</font> ) </span> } </span>  implies ( <font color="Maroon" title="c2">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c3">F</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a>  &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">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</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="c3">F</font> ) ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
( <font color="Maroon" title="c2">M</font> is  <a href="armstrng.html#V8" title="ARMSTRNG:attr.8">(M1)</a>  &amp; <font color="Maroon" title="c2">M</font> is  <a href="armstrng.html#V9" title="ARMSTRNG:attr.9">(M2)</a>  &amp; <font color="Maroon" title="c2">M</font> is  <a href="armstrng.html#V10" title="ARMSTRNG:attr.10">(M3)</a>  )
 <span class="kw">and </span><br/><a NAME="E2:71"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">A</font>,<font color="Olive" title="b2">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b3">A9</font>, <font color="Olive" title="b4">B9</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/>( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">A9</font>,<font color="Olive" title="b4">B9</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="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">A9</font>,<font color="Olive" title="b4">B9</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">M</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c3">F</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a>  &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">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</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="c3">F</font> ) )</span><br/>

<a NAME="E3:71"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="armstrng.html#D11" target="_self" title="ARMSTRNG:def.11">ARMSTRNG:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</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="c3">F</font></span><br/>

<a NAME="E1:71_1"/>
 ex <font color="Olive" title="b1">A9</font>, <font color="Olive" title="b2">B9</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/>( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A9</font>,<font color="Olive" title="b2">B9</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="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A9</font>,<font color="Olive" title="b2">B9</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">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:71_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</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="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E44">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> implies  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</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="Maroon" title="c4">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</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">M</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:71_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</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="Maroon" title="c4">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</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">M</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">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="E3:71_2"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c4">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="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:71_2"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
 ex <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">b9</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/>( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</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="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</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">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">a9</font>, <font color="Maroon" title="c8">b9</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="E6:71_2"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c8">b9</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="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:71_2"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c8">b9</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">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71_2"><span class="lab"><font color="Green" title="E46">A6</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c5">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">a9</font>, <font color="Olive" title="b3">b9</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="Maroon" title="c4">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="c9">a</font>,<font color="Olive" title="b1">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b2">a9</font>,<font color="Olive" title="b3">b9</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="c9">a</font>,<font color="Olive" title="b1">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b2">a9</font>,<font color="Olive" title="b3">b9</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">M</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c6">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">b9</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="Maroon" title="c4">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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</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">M</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c11">a9</font> = <font color="Maroon" title="c7">a9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b9</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="Maroon" title="c4">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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a9</font>,<font color="Olive" title="b1">b9</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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a9</font>,<font color="Olive" title="b1">b9</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">M</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c12">b9</font> = <font color="Maroon" title="c8">b9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">b9</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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">b9</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">M</font> )</span><br/><span class="kw">thus </span><a NAME="E8:71_2"/>
( <font color="Maroon" title="c4">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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">b9</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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">b9</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">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E6:71_2"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E7:71_2"><span class="lab"><font color="Green" title="E48">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E45">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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="c3">F</font> implies  ex <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">b9</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/>( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</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="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</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">M</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:71_3"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">b9</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/>( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</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="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</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">M</font> )</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>, <font color="Maroon" title="c8">a9</font>, <font color="Maroon" title="c9">b9</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="E3:71_3"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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="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">and </span><br/><a NAME="E4:71_3"/><span class="lab"><font color="Green" title="E47">A11</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</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="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">and </span><br/><a NAME="E5:71_3"/><span class="lab"><font color="Green" title="E48">A12</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</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">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E44">A4</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c10">a9</font> = <font color="Maroon" title="c8">a9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b9</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/>( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a9</font>,<font color="Olive" title="b1">b9</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="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a9</font>,<font color="Olive" title="b1">b9</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">M</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c11">b9</font> = <font color="Maroon" title="c9">b9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c11">b9</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="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c11">b9</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">M</font> )</span><br/><span class="kw">thus </span><a NAME="E6:71_3"/>
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c11">b9</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="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c11">b9</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">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_3"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E4:71_3"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="txt" href="#E5:71_3"><span class="lab"><font color="Green" title="E48">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font>, <font color="Maroon" title="c6">C</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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="c3">F</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</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="c3">F</font> implies <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</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="c3">F</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71_4"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E2:71_4"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</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="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</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="c3">F</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c7">A9</font>, <font color="Maroon" title="c8">B9</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="E4:71_4"/><span class="lab"><font color="Green" title="E48">A15</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A9</font>,<font color="Maroon" title="c8">B9</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="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:71_4"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A9</font>,<font color="Maroon" title="c8">B9</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">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E1:71_4"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">B19</font>, <font color="Maroon" title="c10">C9</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="E7:71_4"/><span class="lab"><font color="Green" title="E50">A17</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">B19</font>,<font color="Maroon" title="c10">C9</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="c5">B</font>,<font color="Maroon" title="c6">C</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:71_4"/><span class="lab"><font color="Green" title="E51">A18</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">B19</font>,<font color="Maroon" title="c10">C9</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">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E2:71_4"><span class="lab"><font color="Green" title="E47">A14</font></span></a></span>;<br/><a NAME="E9:71_4"/><span class="lab"><font color="Green" title="E52">A19</font></span>: 
<font color="Maroon" title="c9">B19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71_4"><span class="lab"><font color="Green" title="E50">A17</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/><a NAME="E10:71_4"/>
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71_4"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/><a NAME="E11:71_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">B19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:71_4"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E12:71_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">C9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E5:71_4"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="txt" href="#E8:71_4"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="ref" href="armstrng.html#D20" target="_self" title="ARMSTRNG:def.20">Def21</a></span>;<br/><a NAME="E13:71_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A20</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A9</font>,<font color="Maroon" title="c8">B9</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="c7">A9</font>,<font color="Maroon" title="c10">C9</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#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/><a NAME="E14:71_4"/><span class="lab"><font color="Green" title="E54">A21</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">C9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71_4"><span class="lab"><font color="Green" title="E50">A17</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/><a NAME="E15:71_4"/>
<font color="Maroon" title="c7">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71_4"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/><a NAME="E16:71_4"/><span class="kw">then </span>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A9</font>,<font color="Maroon" title="c10">C9</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="c4">A</font>,<font color="Maroon" title="c6">C</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:71_4"><span class="lab"><font color="Green" title="E54">A21</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/><a NAME="E17:71_4"/><span class="kw">then </span>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A9</font>,<font color="Maroon" title="c8">B9</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="c4">A</font>,<font color="Maroon" title="c6">C</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:71_4"><span class="lab"><font color="Green" title="E53">A20</font></span></a>, <a class="ref" href="armstrng.html#T12" target="_self" title="ARMSTRNG:th.12">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E18:71_4"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</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="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E5:71_4"><span class="lab"><font color="Green" title="E49">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="armstrng.html#NV4" title="ARMSTRNG:NV.4">(F2)</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T18" target="_self" title="ARMSTRNG:th.18">Th20</a></span>;<br/>
<a NAME="E8:71"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font>, <font color="Maroon" title="c6">A9</font>, <font color="Maroon" title="c7">B9</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="armstrng.html#D13" target="_self" title="ARMSTRNG:def.13">ARMSTRNG:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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="c3">F</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">A9</font>,<font color="Maroon" title="c7">B9</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="c3">F</font> implies <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></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="c3">F</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71_5"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E2:71_5"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">A9</font>,<font color="Maroon" title="c7">B9</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="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></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="c3">F</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c9">b1</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="E4:71_5"/><span class="lab"><font color="Green" title="E50">A26</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">b1</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="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:71_5"/><span class="lab"><font color="Green" title="E51">A27</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">b1</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">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E1:71_5"><span class="lab"><font color="Green" title="E48">A24</font></span></a></span>;<br/>
<a NAME="E6:71_5"/><span class="lab"><font color="Green" title="E52">A28</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71_5"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">a19</font>, <font color="Maroon" title="c11">b19</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="E8:71_5"/><span class="lab"><font color="Green" title="E53">A29</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a19</font>,<font color="Maroon" title="c11">b19</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="c6">A9</font>,<font color="Maroon" title="c7">B9</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:71_5"/><span class="lab"><font color="Green" title="E54">A30</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a19</font>,<font color="Maroon" title="c11">b19</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">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E2:71_5"><span class="lab"><font color="Green" title="E49">A25</font></span></a></span>;<br/>
<a NAME="E10:71_5"/><span class="lab"><font color="Green" title="E55">A31</font></span>: 
<font color="Maroon" title="c7">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">b19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_5"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">A99</font>, <font color="Maroon" title="c13">B99</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="E12:71_5"/><span class="lab"><font color="Green" title="E56">A32</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c12">A99</font>,<font color="Maroon" title="c13">B99</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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A9</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E13:71_5"/><span class="lab"><font color="Green" title="E57">A33</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c12">A99</font>,<font color="Maroon" title="c13">B99</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">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def19</a></span>;<br/>
<a NAME="E14:71_5"/><span class="lab"><font color="Green" title="E58">A34</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:71_5"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/>
<a NAME="E15:71_5"/>
<font color="Maroon" title="c10">a19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_5"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/>
<a NAME="E16:71_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">a19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></span>;<br/>
<a NAME="E17:71_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">a19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:71_5"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E18:71_5"/><span class="kw">then </span>
<font color="Maroon" title="c11">b19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E13:71_5"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E9:71_5"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="armstrng.html#D20" target="_self" title="ARMSTRNG:def.20">Def21</a></span>;<br/>
<a NAME="E19:71_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A35</font></span>: 
<font color="Maroon" title="c7">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:71_5"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E20:71_5"/>
<font color="Maroon" title="c8">a1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71_5"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/>
<a NAME="E21:71_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">a1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></span>;<br/>
<a NAME="E22:71_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">a1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:71_5"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E23:71_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E13:71_5"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E5:71_5"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="ref" href="armstrng.html#D20" target="_self" title="ARMSTRNG:def.20">Def21</a></span>;<br/>
<a NAME="E24:71_5"/><span class="kw">then </span>
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_5"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E25:71_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A36</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B99</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c13">B99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:71_5"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></span>;<br/>
<a NAME="E26:71_5"/>
<font color="Maroon" title="c12">A99</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:71_5"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/>
<a NAME="E27:71_5"/><span class="kw">then </span>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c12">A99</font>,<font color="Maroon" title="c13">B99</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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:71_5"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:71_5"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></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="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E13:71_5"><span class="lab"><font color="Green" title="E57">A33</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c4">DOX</font> =  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_6"><span class="kw">now </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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> implies <font color="Maroon" title="c5">x</font> <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="c3">F</font> ) &amp; ( <font color="Maroon" title="c5">x</font> <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="c3">F</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <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="c3">F</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:71_6_1"/><span class="lab"><font color="Green" title="E48">A37</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">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <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="c3">F</font></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="E3:71_6_1"/><span class="lab"><font color="Green" title="E49">A38</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#T9" target="_self" title="ARMSTRNG:th.9">Th10</a></span>;<br/><a NAME="E4:71_6_1"/>
<font color="Maroon" title="c5">x</font> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c3">F</font>, <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_6_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:71_6_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E1:71_6_1"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="txt" href="#E3:71_6_1"><span class="lab"><font color="Green" title="E49">A38</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_4.html#D23" title="WAYBEL_4:def.23">WAYBEL_4:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Olive">b<sub>1</sub></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#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E3:71_6_1_1"/><span class="lab"><font color="Green" title="E50">A39</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E4:71_6_1_1"/><span class="lab"><font color="Green" title="E51">A40</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E5:71_6_1_1"/><span class="lab"><font color="Green" title="E52">A41</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c8">y</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#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">e</font>, <font color="Maroon" title="c10">f</font> being   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E7:71_6_1_1"/><span class="lab"><font color="Green" title="E53">A42</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c10">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:71_6_1_1"/><span class="lab"><font color="Green" title="E54">A43</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c10">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71_6_1_1"><span class="lab"><font color="Green" title="E52">A41</font></span></a></span>;<br/>
<a NAME="E9:71_6_1_1"/><span class="lab"><font color="Green" title="E55">A44</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71_6_1_1"><span class="lab"><font color="Green" title="E53">A42</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">c</font>, <font color="Maroon" title="c12">d</font>, <font color="Maroon" title="c13">c9</font>, <font color="Maroon" title="c14">d9</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="E11:71_6_1_1"/><span class="lab"><font color="Green" title="E56">A45</font></span>: 
<font color="Maroon" title="c8">y</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">c</font>,<font color="Maroon" title="c12">d</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E12:71_6_1_1"/><span class="lab"><font color="Green" title="E57">A46</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">c9</font>,<font color="Maroon" title="c14">d9</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">c</font>,<font color="Maroon" title="c12">d</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E13:71_6_1_1"/><span class="lab"><font color="Green" title="E58">A47</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">c9</font>,<font color="Maroon" title="c14">d9</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">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E3:71_6_1_1"><span class="lab"><font color="Green" title="E50">A39</font></span></a></span>;<br/>
<a NAME="E14:71_6_1_1"/><span class="lab"><font color="Green" title="E59">A48</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71_6_1_1"><span class="lab"><font color="Green" title="E53">A42</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E15:71_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A49</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">c9</font>,<font color="Maroon" title="c14">d9</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="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="txt" href="#E3:71_6_1"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="txt" href="#E8:71_6_1_1"><span class="lab"><font color="Green" title="E54">A43</font></span></a>, <a class="txt" href="#E9:71_6_1_1"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="txt" href="#E11:71_6_1_1"><span class="lab"><font color="Green" title="E56">A45</font></span></a>, <a class="txt" href="#E12:71_6_1_1"><span class="lab"><font color="Green" title="E57">A46</font></span></a>, <a class="ref" href="armstrng.html#T12" target="_self" title="ARMSTRNG:th.12">Th14</a></span>;<br/>
<a NAME="E16:71_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A50</font></span>: 
<font color="Maroon" title="c14">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E1:71_6_1"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="txt" href="#E3:71_6_1"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="txt" href="#E13:71_6_1_1"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="ref" href="armstrng.html#D19" target="_self" title="ARMSTRNG:def.19">Def20</a></span>;<br/>
<a NAME="E17:71_6_1_1"/>
<font color="Maroon" title="c13">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E1:71_6_1"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="txt" href="#E3:71_6_1"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="txt" href="#E13:71_6_1_1"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="txt" href="#E15:71_6_1_1"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="ref" href="armstrng.html#D19" target="_self" title="ARMSTRNG:def.19">Def20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:71_6_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_6_1"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="txt" href="#E4:71_6_1_1"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="txt" href="#E8:71_6_1_1"><span class="lab"><font color="Green" title="E54">A43</font></span></a>, <a class="txt" href="#E14:71_6_1_1"><span class="lab"><font color="Green" title="E59">A48</font></span></a>, <a class="txt" href="#E9:71_6_1_1"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="txt" href="#E11:71_6_1_1"><span class="lab"><font color="Green" title="E56">A45</font></span></a>, <a class="txt" href="#E12:71_6_1_1"><span class="lab"><font color="Green" title="E57">A46</font></span></a>, <a class="txt" href="#E16:71_6_1_1"><span class="lab"><font color="Green" title="E61">A50</font></span></a>, <a class="ref" href="armstrng.html#T11" target="_self" title="ARMSTRNG:th.11">Th13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:71_6_1"/>
<font color="Maroon" title="c5">x</font> <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="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#D1" target="_self" title="ARMSTRNG:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:71_6"/><span class="lab"><font color="Green" title="E48">A51</font></span>: 
<font color="Maroon" title="c5">x</font> <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="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font></span><br/><a NAME="E3:71_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A52</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c3">F</font>, <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#D1" target="_self" title="ARMSTRNG:def.1">Def1</a></span>;<br/><span class="kw">assume </span><a NAME="E4:71_6"/><span class="lab"><font color="Green" title="E50">A53</font></span>: 
 not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">a9</font>, <font color="Maroon" title="c9">b9</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="E6:71_6"/><span class="lab"><font color="Green" title="E51">A54</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">and </span><br/><a NAME="E7:71_6"/><span class="lab"><font color="Green" title="E52">A55</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</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="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">and </span><br/><a NAME="E8:71_6"/><span class="lab"><font color="Green" title="E53">A56</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</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">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E2:71_6"><span class="lab"><font color="Green" title="E48">A51</font></span></a></span>;<br/><a NAME="E9:71_6"/><span class="lab"><font color="Green" title="E54">A57</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><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="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span></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#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71_6"><span class="lab"><font color="Green" title="E52">A55</font></span></a></span>;<br/><a NAME="E10:71_6"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</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="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E8:71_6"><span class="lab"><font color="Green" title="E53">A56</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:71_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_6"><span class="lab"><font color="Green" title="E49">A52</font></span></a>, <a class="txt" href="#E6:71_6"><span class="lab"><font color="Green" title="E51">A54</font></span></a>, <a class="txt" href="#E8:71_6"><span class="lab"><font color="Green" title="E53">A56</font></span></a>, <a class="txt" href="#E4:71_6"><span class="lab"><font color="Green" title="E50">A53</font></span></a>, <a class="txt" href="#E9:71_6"><span class="lab"><font color="Green" title="E54">A57</font></span></a>, <a class="ref" href="waybel_4.html#D23" title="WAYBEL_4:def.23">WAYBEL_4:def 23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:71"/>
<font color="Maroon" title="c2">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a>  &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">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</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="c3">F</font> ) )</span><br/>

<a NAME="E11:71"/>
<font color="Maroon" title="c3">F</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font>, <font color="Maroon" title="c7">A9</font>, <font color="Maroon" title="c8">B9</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">ARMSTRNG:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">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="c3">F</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">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="c7">A9</font>,<font color="Maroon" title="c8">B9</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> implies <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A9</font>,<font color="Maroon" title="c8">B9</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="c3">F</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71_7"/><span class="lab"><font color="Green" title="E48">A58</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">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="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E2:71_7"/><span class="lab"><font color="Green" title="E49">A59</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">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="c7">A9</font>,<font color="Maroon" title="c8">B9</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A9</font>,<font color="Maroon" title="c8">B9</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="c3">F</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">a9</font>, <font color="Maroon" title="c10">b9</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="E4:71_7"/><span class="lab"><font color="Green" title="E50">A60</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</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="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:71_7"/><span class="lab"><font color="Green" title="E51">A61</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</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">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E1:71_7"><span class="lab"><font color="Green" title="E48">A58</font></span></a></span>;<br/>
<a NAME="E6:71_7"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</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="c7">A9</font>,<font color="Maroon" title="c8">B9</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_7"><span class="lab"><font color="Green" title="E49">A59</font></span></a>, <a class="txt" href="#E4:71_7"><span class="lab"><font color="Green" title="E50">A60</font></span></a>, <a class="ref" href="armstrng.html#T12" target="_self" title="ARMSTRNG:th.12">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:71_7"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A9</font>,<font color="Maroon" title="c8">B9</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="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E5:71_7"><span class="lab"><font color="Green" title="E51">A61</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:71"/>
<font color="Maroon" title="c3">F</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E7:71"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E8:71"><span class="lab"><font color="Green" title="E47">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</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="c3">F</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c5">G</font> implies <font color="Maroon" title="c5">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E13:71"/><span class="lab"><font color="Green" title="E48">A62</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">F</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> )</span>
<div class="add"><a NAME="E1:71_8_1"/><span class="lab"><font color="Green" title="E49">A63</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T17" target="_self" title="ARMSTRNG:th.17">Th19</a></span>;<br/><span class="kw">assume </span><a NAME="E2:71_8_1"/><span class="lab"><font color="Green" title="E50">A64</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">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="E4:71_8_1"/><span class="lab"><font color="Green" title="E51">A65</font></span>: 
<font color="Maroon" title="c6">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="c7">a</font>,<font color="Maroon" title="c8">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#T9" target="_self" title="ARMSTRNG:th.9">Th10</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">y</font> being   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <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="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> );<br/><span class="kw">consider </span><font color="Maroon" title="c9">MA</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:71_8_1"/><span class="lab"><font color="Green" title="E52">A66</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">MA</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span><a NAME="E7:71_8_1"/>
<font color="Maroon" title="c9">MA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_8_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">MA</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:71_8_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">MA</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font></span><br/>

<span class="kw">hence </span><a NAME="E2:71_8_1_1"/>
<font color="Maroon" title="c10">x</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="#E6:71_8_1"><span class="lab"><font color="Green" title="E52">A66</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:71_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A67</font></span>: 
<font color="Maroon" title="c9">MA</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p2">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_8_1"><span class="lab"><font color="Green" title="E49">A63</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E9:71_8_1"/>
<font color="Maroon" title="c9">MA</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_8_1"><span class="lab"><font color="Green" title="E50">A64</font></span></a>, <a class="txt" href="#E4:71_8_1"><span class="lab"><font color="Green" title="E51">A65</font></span></a>, <a class="txt" href="#E6:71_8_1"><span class="lab"><font color="Green" title="E52">A66</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">MA</font><span class="kw"> such that </span><br/><a NAME="E11:71_8_1"/><span class="lab"><font color="Green" title="E54">A68</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c9">MA</font>, <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_8_1"><span class="lab"><font color="Green" title="E53">A67</font></span></a>, <a class="ref" href="armstrng.html#T2" target="_self" title="ARMSTRNG:th.2">Th2</a></span>;<br/><a NAME="E12:71_8_1"/><span class="lab"><font color="Green" title="E55">A69</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">MA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:71_8_1"><span class="lab"><font color="Green" title="E54">A68</font></span></a>, <a class="ref" href="waybel_4.html#D23" title="WAYBEL_4:def.23">WAYBEL_4:def 23</a></span>;<br/><a NAME="E13:71_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">m</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="#E6:71_8_1"><span class="lab"><font color="Green" title="E52">A66</font></span></a></span>;<br/><a NAME="E14:71_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A70</font></span>: 
 ex <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">b9</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c10">m</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="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</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#T9" target="_self" title="ARMSTRNG:th.9">Th10</a></span>;<br/><a NAME="E15:71_8_1"/>
<font color="Maroon" title="c10">m</font> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c5">G</font>, <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_8_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:71_8_1_2"/><span class="lab"><font color="Green" title="E57">A71</font></span>: 
 ex <font color="Olive" title="b1">mm</font> being   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">mm</font> &amp; <font color="Olive" title="b1">mm</font> <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="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_8_1"><span class="lab"><font color="Green" title="E52">A66</font></span></a>, <a class="txt" href="#E12:71_8_1"><span class="lab"><font color="Green" title="E55">A69</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:71_8_1_2"/>
<font color="Maroon" title="c10">m</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="#E6:71_8_1"><span class="lab"><font color="Green" title="E52">A66</font></span></a>, <a class="txt" href="#E12:71_8_1"><span class="lab"><font color="Green" title="E55">A69</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_4.html#D23" title="WAYBEL_4:def.23">WAYBEL_4:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">m</font>,<font color="Olive">b<sub>1</sub></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#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E4:71_8_1_2"/><span class="lab"><font color="Green" title="E58">A72</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E5:71_8_1_2"/><span class="lab"><font color="Green" title="E59">A73</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">and </span><br/><a NAME="E6:71_8_1_2"/><span class="lab"><font color="Green" title="E60">A74</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">m</font>,<font color="Maroon" title="c11">y</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#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">e</font>, <font color="Maroon" title="c13">f</font> being   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E8:71_8_1_2"/><span class="lab"><font color="Green" title="E61">A75</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">m</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">e</font>,<font color="Maroon" title="c13">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:71_8_1_2"/><span class="lab"><font color="Green" title="E62">A76</font></span>: 
<font color="Maroon" title="c12">e</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c13">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_8_1_2"><span class="lab"><font color="Green" title="E60">A74</font></span></a></span>;<br/>
<a NAME="E10:71_8_1_2"/><span class="lab"><font color="Green" title="E63">A77</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_8_1_2"><span class="lab"><font color="Green" title="E61">A75</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E11:71_8_1_2"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_8_1_2"><span class="lab"><font color="Green" title="E61">A75</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E12:71_8_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">f</font> <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="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_8_1_2"><span class="lab"><font color="Green" title="E57">A71</font></span></a>, <a class="txt" href="#E9:71_8_1_2"><span class="lab"><font color="Green" title="E62">A76</font></span></a>, <a class="ref" href="armstrng.html#T12" target="_self" title="ARMSTRNG:th.12">Th14</a></span>;<br/>
<a NAME="E13:71_8_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">MA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_8_1"><span class="lab"><font color="Green" title="E52">A66</font></span></a>, <a class="txt" href="#E4:71_8_1_2"><span class="lab"><font color="Green" title="E58">A72</font></span></a>, <a class="txt" href="#E10:71_8_1_2"><span class="lab"><font color="Green" title="E63">A77</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:71_8_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:71_8_1"><span class="lab"><font color="Green" title="E54">A68</font></span></a>, <a class="txt" href="#E5:71_8_1_2"><span class="lab"><font color="Green" title="E59">A73</font></span></a>, <a class="txt" href="#E6:71_8_1_2"><span class="lab"><font color="Green" title="E60">A74</font></span></a>, <a class="ref" href="waybel_4.html#D23" title="WAYBEL_4:def.23">WAYBEL_4:def 23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E16:71_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A78</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="armstrng.html#K1" title="ARMSTRNG:func.1">Maximal_in</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#D1" target="_self" title="ARMSTRNG:def.1">Def1</a></span>;<br/><a NAME="E17:71_8_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <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="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_8_1"><span class="lab"><font color="Green" title="E52">A66</font></span></a>, <a class="txt" href="#E12:71_8_1"><span class="lab"><font color="Green" title="E55">A69</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:71_8_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E13:71"><span class="lab"><font color="Green" title="E48">A62</font></span></a>, <a class="txt" href="#E4:71_8_1"><span class="lab"><font color="Green" title="E51">A65</font></span></a>, <a class="txt" href="#E16:71_8_1"><span class="lab"><font color="Green" title="E57">A78</font></span></a>, <a class="txt" href="#E14:71_8_1"><span class="lab"><font color="Green" title="E56">A70</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:71_8"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font></span><br/><a NAME="E3:71_8"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">b1</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="Maroon" title="c6">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">a1</font>,<font color="Olive" title="b4">b1</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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">a1</font>,<font color="Olive" title="b4">b1</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">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E44">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:71_8"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:71"><span class="lab"><font color="Green" title="E48">A62</font></span></a>, <a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:71"/>
<font color="Maroon" title="c5">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
