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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:196"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:196"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E3:196"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E4:196"/>
<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="E5:196"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:196"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="orders_1.html#T151" target="_self" title="ORDERS_1:th.151">Th151</a></span>;<br/>
<a NAME="E6:196"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:196"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a></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="E8:196"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:196"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E5:196"><span class="lab"><font color="Green" title="E50">A4</font></span></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#T173" target="_self" title="ORDERS_1:th.173">Th173</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">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">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/>

<span class="kw">thus </span><a NAME="E9:196"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E8:196"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="orders_1.html#T163" target="_self" title="ORDERS_1:th.163">Th163</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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