<?xml version="1.0"?>
<div class="add">

<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 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="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">B</font> is  <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a>  &amp; <font color="Olive" title="b1">B</font> is  <a href="armstrng.html#NV2" title="ARMSTRNG:NV.2">(B2)</a>  holds <br/> <a href="armstrng.html#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Olive" title="b1">B</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a> <font color="Olive" title="b1">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</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="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"> ( <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a>  &amp; <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#NV2" title="ARMSTRNG:NV.2">(B2)</a>  implies  <a href="armstrng.html#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a> <font color="Maroon" title="c2">B</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:96"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
( <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a>  &amp; <font color="Maroon" title="c2">B</font> is  <a href="armstrng.html#NV2" title="ARMSTRNG:NV.2">(B2)</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#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a> <font color="Maroon" title="c2">B</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">G</font> =  <a href="armstrng.html#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font>;<br/>
<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#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font> </span>}</span> ;<br/>
<span class="kw">thus </span><a NAME="E2:96"/>
 <a href="armstrng.html#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</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"> <font color="Maroon" title="c2">B</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#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font> </span>}</span> </span><br/>

<a NAME="E3:96"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_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="c2">B</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#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</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#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font> </span>}</span>  implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</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#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font> </span>}</span>  implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:96_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"> <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#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font> </span>}</span> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">xx</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">B</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c7">yz</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">B</font><span class="kw"> such that </span><br/><a NAME="E4:96_1_1"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
<font color="Maroon" title="c6">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">yz</font>
 <span class="kw">and </span><br/><a NAME="E5:96_1_1"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">yz</font> holds <br/><font color="Olive" title="b1">s</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T3" target="_self" title="ARMSTRNG:th.3">Th3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">yz</font> = <font color="Maroon" title="c7">yz</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E7:96_1_1"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
<font color="Maroon" title="c8">yz</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1"><span class="kw">proof </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> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">yz</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">yz</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font></span><br/>

<a NAME="E2:96_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_1_1"><span class="lab"><font color="Green" title="E57">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:96_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#D3" target="_self" title="ARMSTRNG:def.3">Def3</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="E8:96_1_1"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c8">yz</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="c8">yz</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/><span class="kw">hence </span><a NAME="E9:96_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#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_1_1"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E7:96_1_1"><span class="lab"><font color="Green" title="E58">A5</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">assume </span><a NAME="E2:96_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#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</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> <font color="Maroon" title="c2">B</font></span><br/><a NAME="E3:96_1"/><span class="kw">then </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/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</font> )
 ;<br/><span class="kw">hence </span><a NAME="E4:96_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="armstrng.html#T1" target="_self" title="ARMSTRNG:th.1">Th1</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="E5:96"/>
<font color="Maroon" title="c2">B</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#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c2">B</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>
