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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">G</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>  &amp; <font color="Maroon" title="c2">G</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a> <font color="Olive" title="b1">B</font> holds <br/> <a href="armstrng.html#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="c3">B</font> is  <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a>  &amp; <font color="Maroon" title="c3">B</font> is  <a href="armstrng.html#NV2" title="ARMSTRNG:NV.2">(B2)</a>  &amp; <font color="Maroon" title="c2">G</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a> <font color="Maroon" title="c3">B</font> implies  <a href="armstrng.html#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:97"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
( <font color="Maroon" title="c3">B</font> is  <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a>  &amp; <font color="Maroon" title="c3">B</font> is  <a href="armstrng.html#NV2" title="ARMSTRNG:NV.2">(B2)</a>  )
 <span class="kw">and </span><br/><a NAME="E2:97"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a> <font color="Maroon" title="c3">B</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#K2" title="ARMSTRNG:func.2">/\-IRR</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E3:97"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
<font color="Maroon" title="c3">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> <font color="Maroon" title="c2">G</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="armstrng.html#D26" target="_self" title="ARMSTRNG:def.26">Def26</a></span>;<br/>
<a NAME="E4:97"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="armstrng.html#D26" target="_self" title="ARMSTRNG:def.26">Def26</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">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="c4">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="c3">B</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E5:97"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
<font color="Maroon" title="c4">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="c3">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">xx</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">B</font> ;<br/>
<a NAME="E7:97"/><span class="lab"><font color="Green" title="E59">A6</font></span>: 
<font color="Maroon" title="c5">xx</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="armstrng.html#D3" target="_self" title="ARMSTRNG:def.3">Def3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:97"/><span class="lab"><font color="Green" title="E60">A7</font></span>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E9:97"/><span class="kw">then </span>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E10:97"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E11:97"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E58">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">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><span class="kw"> such that </span><br/><a NAME="E13:97"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
<font color="Maroon" title="c4">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="Maroon" title="c6">S</font>
 <span class="kw">and </span><br/><a NAME="E14:97"/><span class="lab"><font color="Green" title="E63">A10</font></span>: 
<font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E56">A3</font></span></a></span>;<br/>
<a NAME="E15:97"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:97"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E8:97"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:97"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E13:97"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="txt" href="#E14:97"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E10:97"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E7:97"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="ref" href="armstrng.html#T4" target="_self" title="ARMSTRNG:th.4">Th4</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/>


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