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<span class="kw">let </span><font color="Maroon" title="c1">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font><br/> for <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> holds  <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Olive" title="b1">B</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a> </span><br/><span class="kw">let </span><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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a> </span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_1"/>
(  not <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#NV15" title="ARMSTRNG:NV.15">(C1)</a>  or <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#NV15" title="ARMSTRNG:NV.15">(C1)</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><span class="kw">suppose </span></a><a NAME="E1:82_1_1"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
not <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#NV15" title="ARMSTRNG:NV.15">(C1)</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a> </span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_1"><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="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> ) &amp; ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:82_1_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> )</span><br/><span class="kw">assume </span><a NAME="E2:82_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:82_1_1_1"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E5:82_1_1_1"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E6:82_1_1_1"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<font color="Maroon" title="c5">x</font> <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="c6">x1</font>,<font color="Maroon" title="c7">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/><a NAME="E7:82_1_1_1"/>
for <font color="Olive" title="b1">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Maroon" title="c7">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:82_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_1_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E5:82_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E6:82_1_1_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:82_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/><a NAME="E4:82_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#NV4" title="ARMSTRNG:NV.4">(F2)</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T21" target="_self" title="ARMSTRNG:th.21">Th21</a></span>;<br/><span class="kw">hence </span><a NAME="E5:82_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><span class="kw">suppose </span></a><a NAME="E1:82_1_2"/>
<font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#NV15" title="ARMSTRNG:NV.15">(C1)</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c2">B</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/><span class="kw">thus </span><a NAME="E3:82_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a> 
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="armstrng.html#D15" target="_self" title="ARMSTRNG:def.15">ARMSTRNG:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#NV4" title="ARMSTRNG:NV.4">(F2)</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="Red">:: 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="Red">::  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="c6">A</font>,<font color="Maroon" title="c6">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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font></span><br/>

<a NAME="E1:82_1_2_1"/>
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E2:82_1_2_1"/>
<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="c6">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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">B</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">c</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font> implies <font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82_1_2_2"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E2:82_1_2_2"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E4:82_1_2_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="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:82_1_2_2"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c9">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c10">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_2_2"><span class="lab"><font color="Green" title="E46">A6</font></span></a></span>;<br/><a NAME="E6:82_1_2_2"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2_2"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E7:82_1_2_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2_2"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E8:82_1_2_2"/>
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_2_2"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E5:82_1_2_2"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E6:82_1_2_2"><span class="lab"><font color="Green" title="E50">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">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="Red">::  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="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> implies <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="c8">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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82_1_2_3"/><span class="lab"><font color="Green" title="E46">A11</font></span>: 
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E2:82_1_2_3"/><span class="lab"><font color="Green" title="E47">A12</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c6">a</font>,<font color="Maroon" title="c8">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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">x</font> implies <font color="Maroon" title="c8">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">x</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82_1_2_3_1"/><span class="lab"><font color="Green" title="E48">A13</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">and </span><br/><a NAME="E2:82_1_2_3_1"/><span class="lab"><font color="Green" title="E49">A14</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">x</font></span><br/><a NAME="E3:82_1_2_3_1"/>
<font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E1:82_1_2_3"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E1:82_1_2_3_1"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="txt" href="#E2:82_1_2_3_1"><span class="lab"><font color="Green" title="E49">A14</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:82_1_2_3_1"/>
<font color="Maroon" title="c8">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E2:82_1_2_3"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="txt" href="#E1:82_1_2_3_1"><span class="lab"><font color="Green" title="E48">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:82_1_2_3"/>
<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="c8">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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:82_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> 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#T20" target="_self" title="ARMSTRNG:th.20">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>  )</span><br/><span class="kw">thus </span><a NAME="E7:82_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </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> 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="Red">:: 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="Red">::  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="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> &amp; <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> <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="c8">a9</font>,<font color="Maroon" title="c9">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="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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82_1_2_4"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E2:82_1_2_4"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<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> <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="c8">a9</font>,<font color="Maroon" title="c9">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="Red">::  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="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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font></span><br/>

<a NAME="E3:82_1_2_4"/><span class="lab"><font color="Green" title="E48">A17</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_2_4"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></span>;<br/>
<a NAME="E4:82_1_2_4"/><span class="lab"><font color="Green" title="E49">A18</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_2_4"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c8">a9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82_1_2_4_1"/><span class="lab"><font color="Green" title="E50">A19</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">and </span><br/><a NAME="E2:82_1_2_4_1"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
<font color="Maroon" title="c8">a9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font></span><br/><a NAME="E3:82_1_2_4_1"/>
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2_4"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:82_1_2_4_1"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E4:82_1_2_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E1:82_1_2_4"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E1:82_1_2_4_1"><span class="lab"><font color="Green" title="E50">A19</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:82_1_2_4_1"/>
<font color="Maroon" title="c9">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1_2_4"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:82_1_2_4"/>
<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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E8:82_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </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> 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="Red">:: according to </font></span><a class="ref" href="armstrng.html#D14" target="_self" title="ARMSTRNG:def.14">ARMSTRNG:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> &amp; <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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</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="c6">a</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">a9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82_1_2_5"/><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E2:82_1_2_5"/><span class="lab"><font color="Green" title="E47">A22</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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c6">a</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">a9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c6">a</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">a9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c7">b</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82_1_2_5_1"/><span class="lab"><font color="Green" title="E48">A23</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">and </span><br/><a NAME="E2:82_1_2_5_1"/><span class="lab"><font color="Green" title="E49">A24</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">a9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font></span><br/><a NAME="E3:82_1_2_5_1"/>
<font color="Maroon" title="c8">a9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_2_5_1"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a></span>;<br/><a NAME="E4:82_1_2_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A25</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E2:82_1_2_5"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="txt" href="#E1:82_1_2_5_1"><span class="lab"><font color="Green" title="E48">A23</font></span></a></span>;<br/><a NAME="E5:82_1_2_5_1"/>
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_2_5_1"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a></span>;<br/><a NAME="E6:82_1_2_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E1:82_1_2_5"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="txt" href="#E1:82_1_2_5_1"><span class="lab"><font color="Green" title="E48">A23</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:82_1_2_5_1"/>
<font color="Maroon" title="c7">b</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2_5_1"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:82_1_2_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="c6">a</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">a9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">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="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

</div>
