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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="armstrng.html#NM5" title="ARMSTRNG:NM.5">Full-family</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">G</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b1">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Olive" title="b2">G</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</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="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Olive" title="b1">G</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">G</font> =  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>;<br/>
<span class="kw">take </span>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</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"> (  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font></span>)</span> )</span><br/>

<a NAME="E1:95"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
(  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a>  &amp;  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#NV2" title="ARMSTRNG:NV.2">(B2)</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T34" target="_self" title="ARMSTRNG:th.34">Th34</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:95"/>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</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="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c4">H</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> ;<br/>
<span class="kw">thus </span><a NAME="E1:95_1"/>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="armstrng.html#D26" target="_self" title="ARMSTRNG:def.26">ARMSTRNG:def 26</a> <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#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_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>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span>  ) &amp; ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span>  implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span>  implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> )</span>
<div class="add"><span class="kw">set </span><font color="Maroon" title="c6">sx</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><span class="kw">assume </span><a NAME="E1:95_1_1_1"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> </span><br/><a NAME="E2:95_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A3</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">sx</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_1"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E4:95_1_1_1"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">sx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">sx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></span>;<br/><a NAME="E5:95_1_1_1"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">sx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:95_1_1_1"><span class="lab"><font color="Green" title="E56">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:95_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:95_1_1_1"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</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="E2:95_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font></span><br/><a NAME="E3:95_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A5</font></span>: 
 ex <font color="Olive" title="b1">S</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> )
 ;<br/><a NAME="E4:95_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="armstrng.html#D4" target="_self" title="ARMSTRNG:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:95_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E3:95_1_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="armstrng.html#T1" target="_self" title="ARMSTRNG:th.1">Th1</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="E3:95_1"/>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 <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/>


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

<span class="kw">thus </span><a NAME="E3:95"/>
<font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="armstrng.html#T37" target="_self" title="ARMSTRNG:th.37">Th37</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>
