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<b>let </b><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> ;<br/><b>let </b><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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:97"/><i><font color="Green" title="E55">A1</font></i>: 
( <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="finsub_1.html#V2" title="FINSUB_1:attr.2">(B2)</a> )
 <b>and </b><br/><a NAME="E2:97"/><i><font color="Green" title="E56">A2</font></i>: 
<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>
 ;<br/>

<a NAME="E3:97"/><i><font color="Green" title="E57">A3</font></i>: 
<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> 
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:97"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="armstrng.html#D26" target="_self" title="ARMSTRNG:def.26">Def26</a></i>;<br/>
<a NAME="E4:97"/><i><font color="Green" title="E58">A4</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:97"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="armstrng.html#D26" target="_self" title="ARMSTRNG:def.26">Def26</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:97"/><i><font color="Green" title="E59">A5</font></i>: 
<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>
 ;<br/>

<a NAME="E6:97"/><b>then </b>
<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>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c5">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><b> such that </b><br/><a NAME="E8:97"/><i><font color="Green" title="E60">A6</font></i>: 
<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="c5">S</font>
 <b>and </b><br/><a NAME="E9:97"/><i><font color="Green" title="E61">A7</font></i>: 
<font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:97"><i><font color="Green" title="E57">A3</font></i></a></i>;<br/>
<a NAME="E10:97"/>
<font color="Maroon" title="c2">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:97"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E11:97"/><b>then </b><i><font color="Green" title="E62">A8</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="armstrng.html#E9:97"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<b>assume </b><a NAME="E12:97"/><i><font color="Green" title="E63">A9</font></i>: 
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>
 ;<br/>

<a NAME="E13:97"/><b>then </b>
( 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> &amp; 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> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E14:97"/><b>then </b><i><font color="Green" title="E64">A10</font></i>: 
( 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> &amp; <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> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:97"/><i><font color="Green" title="E65">A11</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E9:97"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="armstrng.html#E12:97"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">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> <b>by </b><i><a class="txt" href="armstrng.html#E5:97"><i><font color="Green" title="E59">A5</font></i></a></i>;<br/>
<a NAME="E17:97"/>
<font color="Maroon" title="c6">xx</font> <a href="armstrng.html#R1" title="ARMSTRNG:pred.1">is_/\-irreducible_in</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E5:97"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="armstrng.html#D3" target="_self" title="ARMSTRNG:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E18:97"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E1:97"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="armstrng.html#E4:97"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="armstrng.html#E8:97"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="armstrng.html#E9:97"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="armstrng.html#E11:97"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="armstrng.html#E14:97"><i><font color="Green" title="E64">A10</font></i></a>, <a class="txt" href="armstrng.html#E15:97"><i><font color="Green" title="E65">A11</font></i></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></i>;<br/>


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