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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being  <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">A</font>  st  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Olive" title="b2">G</font> holds <br/> <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Olive" title="b1">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Olive" title="b2">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be   <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c3">G</font> implies  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:116"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
 <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">DCF</font> =  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">DCG</font> =  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><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="Red">::  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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font> ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font> ) )</span><br/><div><span class="kw">hereby </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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:116_1_1"/><span class="lab"><font color="Green" title="E62">A2</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</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">A</font><span class="kw"> such that </span><br/><a NAME="E3:116_1_1"/><span class="lab"><font color="Green" title="E63">A3</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#T10" target="_self" title="ARMSTRNG:th.10">Th10</a></span>;<br/><a NAME="E4:116_1_1"/>
for <font color="Olive" title="b1">c</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>  st <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> &amp; not <font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E1:116_1_1"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="txt" href="#E3:116_1_1"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="armstrng.html#T59" target="_self" title="ARMSTRNG:th.59">Th59</a></span>;<br/><span class="kw">hence </span><a NAME="E5:116_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_1_1"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="armstrng.html#T59" target="_self" title="ARMSTRNG:th.59">Th59</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">assume </span><a NAME="E2:116_1"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">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">A</font><span class="kw"> such that </span><br/><a NAME="E4:116_1"/><span class="lab"><font color="Green" title="E63">A5</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#T10" target="_self" title="ARMSTRNG:th.10">Th10</a></span>;<br/><a NAME="E5:116_1"/>
for <font color="Olive" title="b1">c</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>  st <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> &amp; not <font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E2:116_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E4:116_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="armstrng.html#T59" target="_self" title="ARMSTRNG:th.59">Th59</a></span>;<br/><span class="kw">hence </span><a NAME="E6:116_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="armstrng.html#T59" target="_self" title="ARMSTRNG:th.59">Th59</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="E3:116"/>
 <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</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/>


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