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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>



<a NAME="E1:81"/><i><font color="Green" title="E49">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></i>;<br/>
<b>thus </b><a NAME="E2:81"/>
( <font color="Maroon" title="c2">a</font> <a href="orders_1.html#R7" title="ORDERS_1:pred.7">is_minimal_in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> implies for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  holds not <font color="Olive" title="b1">b</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1"/><i><font color="Green" title="E50">A2</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="orders_1.html#R7" title="ORDERS_1:pred.7">is_minimal_in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>

<a NAME="E2:81_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or  not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:81"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="orders_2.html#E1:81_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="orders_1.html#D12" title="ORDERS_1:def.12">ORDERS_1:def 12</a></i>;<br/>
<a NAME="E3:81_1"/><b>then </b>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or  not <font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" target="_self" title="ORDERS_2:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E4:81_1"/>
not <font color="Maroon" title="c3">b</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D10" target="_self" title="ORDERS_2:def.10">Def10</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:81"/><i><font color="Green" title="E50">A3</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  holds not <font color="Olive" title="b1">b</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<b>thus </b><a NAME="E4:81"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:81"><i><font color="Green" title="E49">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D12" title="ORDERS_1:def.12">ORDERS_1:def 12</a><br/>

<b>let </b><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E5:81"/><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E6:81"/><i><font color="Green" title="E52">A5</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 <b>and </b><br/><a NAME="E7:81"/><i><font color="Green" title="E53">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="orders_2.html#E5:81"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="orders_1.html#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></i>;<br/>
<a NAME="E9:81"/>
<font color="Maroon" title="c4">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E7:81"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="orders_2.html#D9" target="_self" title="ORDERS_2:def.9">Def9</a></i>;<br/>
<a NAME="E10:81"/><b>then </b>
<font color="Maroon" title="c4">b</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E6:81"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="orders_2.html#D10" target="_self" title="ORDERS_2:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E11:81"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_2.html#E3:81"><i><font color="Green" title="E50">A3</font></i></a></i>;<br/>


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