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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c1">R</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Olive" title="b1">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="orders_1.html#R5" title="ORDERS_1:pred.5">has_lower_Zorn_property_wrt</a> <font color="Maroon" title="c1">R</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b2">x</font> <a href="orders_1.html#R7" title="ORDERS_1:pred.7">is_minimal_in</a> <font color="Maroon" title="c1">R</font></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c2">X</font> <a href="orders_1.html#R5" title="ORDERS_1:pred.5">has_lower_Zorn_property_wrt</a> <font color="Maroon" title="c1">R</font> implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">x</font> <a href="orders_1.html#R7" title="ORDERS_1:pred.7">is_minimal_in</a> <font color="Maroon" title="c1">R</font> )</span><br/>



<b>assume </b><a NAME="E1:86"/><i><font color="Green" title="E50">A1</font></i>: 
( <font color="Maroon" title="c1">R</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c2">X</font> <a href="orders_1.html#R5" title="ORDERS_1:pred.5">has_lower_Zorn_property_wrt</a> <font color="Maroon" title="c1">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">x</font> <a href="orders_1.html#R7" title="ORDERS_1:pred.7">is_minimal_in</a> <font color="Maroon" title="c1">R</font></span><br/>

<a NAME="E2:86"/>
<font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 
;<br/>
<a NAME="E3:86"/><b>then </b>
( <font color="Maroon" title="c2">X</font> <a href="orders_1.html#R4" title="ORDERS_1:pred.4">has_upper_Zorn_property_wrt</a> <font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  &amp; <font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E1:86"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="orders_1.html#T137" target="_self" title="ORDERS_1:th.137">Th137</a>, <a class="ref" href="orders_1.html#T151" target="_self" title="ORDERS_1:th.151">Th151</a>, <a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:86"/><i><font color="Green" title="E51">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="orders_1.html#R6" title="ORDERS_1:pred.6">is_maximal_in</a> <font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 <b>by </b><i><a class="ref" href="orders_1.html#T173" target="_self" title="ORDERS_1:th.173">Th173</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="orders_1.html#R7" title="ORDERS_1:pred.7">is_minimal_in</a> <font color="Maroon" title="c1">R</font></span><br/>

<b>thus </b><a NAME="E6:86"/>
<font color="Maroon" title="c3">x</font> <a href="orders_1.html#R7" title="ORDERS_1:pred.7">is_minimal_in</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E5:86"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="orders_1.html#T163" target="_self" title="ORDERS_1:th.163">Th163</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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