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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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">A</font> being  <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c1">R</font>  st not  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> holds <br/> not  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="dilworth.html#NM2" title="DILWORTH:NM.2">StableSet</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> implies  not  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:60"/>
not  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:60"/><span class="lab"><font color="Green" title="E31">B</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E4:60"/><span class="lab"><font color="Green" title="E32">C</font></span>: 
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:60"/><span class="lab"><font color="Green" title="E33">D</font></span>: 
 <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dilworth.html#K4" title="DILWORTH:func.4">Upper</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">x9</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E31">B</font></span></a></span>;<br/>
<a NAME="E7:60"/>
not <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E31">B</font></span></a></span>;<br/>
<a NAME="E8:60"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">B1</font></span>: 
<font color="Maroon" title="c4">x9</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E31">B</font></span></a>, <a class="ref" href="dilworth.html#D9" target="_self" title="DILWORTH:def.9">Lmin</a></span>;<br/>
<a NAME="E9:60"/>
<font color="Maroon" title="c4">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60"><span class="lab"><font color="Green" title="E32">C</font></span></a>, <a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E31">B</font></span></a>, <a class="txt" href="#E5:60"><span class="lab"><font color="Green" title="E33">D</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">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x99</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E11:60"/><span class="lab"><font color="Green" title="E35">C1</font></span>: 
<font color="Maroon" title="c5">x99</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x9</font>
 <span class="kw">and </span><br/><a NAME="E12:60"/><span class="lab"><font color="Green" title="E36">D1</font></span>: 
<font color="Maroon" title="c5">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:60_1"/>
<font color="Maroon" title="c5">x99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x9</font>
 ; <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="E2:60_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x99</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:60"><span class="lab"><font color="Green" title="E35">C1</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:60_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E31">B</font></span></a>, <a class="txt" href="#E8:60"><span class="lab"><font color="Green" title="E34">B1</font></span></a>, <a class="ref" href="waybel_4.html#T57" title="WAYBEL_4:th.57">WAYBEL_4:57</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="E14:60"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:60"><span class="lab"><font color="Green" title="E36">D1</font></span></a>, <a class="txt" href="#E4:60"><span class="lab"><font color="Green" title="E32">C</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/>


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