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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:88"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="orders_1.html#R5" target="_self">has_lower_Zorn_property_wrt</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:88"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K4">~</a> </span>)</span> <a href="relat_1.html#K4">~</a> 
 
;<br/>
<a NAME="E3:88"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_1.html#R4" target="_self">has_upper_Zorn_property_wrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K4">~</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K4">~</a>  <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K4">~</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E1:88"><i><font color="Green">E51</font></i></a>, <a class="ref" href="orders_1.html#T137" target="_self">Th137</a>, <a class="ref" href="orders_1.html#T151" target="_self">Th151</a>, <a class="ref" href="relat_1.html#T38">RELAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:88"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="orders_1.html#R6" target="_self">is_maximal_in</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K4">~</a> 
 <b>by </b><i><a class="ref" href="orders_1.html#T173" target="_self">Th173</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E6:88"/>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_1.html#R7" target="_self">is_minimal_in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="orders_1.html#E5:88"><i><font color="Green">E52</font></i></a>, <a class="ref" href="orders_1.html#T163" target="_self">Th163</a></i>;<br/>


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