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<span class="kw">set </span><font color="Maroon" title="c2">mR</font> =  <a href="dilworth.html#K5" title="DILWORTH:func.5">minimals</a> <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">cR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="dilworth.html#D2" target="_self" title="DILWORTH:def.2">DILWORTH:def 2</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="c4">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> &amp; <font color="Maroon" title="c5">y</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> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font> implies ( not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font> &amp; not <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59_1_1"/><span class="lab"><font color="Green" title="E30">A</font></span>: 
<font color="Maroon" title="c4">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="E2:59_1_1"/><span class="lab"><font color="Green" title="E31">B</font></span>: 
<font color="Maroon" title="c5">y</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="E3:59_1_1"/><span class="lab"><font color="Green" title="E32">C</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font> &amp; not <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> )</span><br/>

<a NAME="E4:59_1_1"/><span class="lab"><font color="Green" title="E33">D</font></span>: 
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="#E1:59_1_1"><span class="lab"><font color="Green" title="E30">A</font></span></a></span>;<br/>
<a NAME="E5:59_1_1"/>
<font color="Maroon" title="c5">y</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="#E2:59_1_1"><span class="lab"><font color="Green" title="E31">B</font></span></a>, <a class="txt" href="#E4:59_1_1"><span class="lab"><font color="Green" title="E33">D</font></span></a>, <a class="ref" href="dilworth.html#D9" target="_self" title="DILWORTH:def.9">Lmin</a></span>;<br/>
<a NAME="E6:59_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c5">y</font> <a href="orders_2.html#NR4" title="ORDERS_2:NR.4">&gt;</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_1"><span class="lab"><font color="Green" title="E30">A</font></span></a>, <a class="ref" href="waybel_4.html#T57" title="WAYBEL_4:th.57">WAYBEL_4:57</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:59_1_1"/>
not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E32">C</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font></span><br/>

<a NAME="E8:59_1_1"/>
<font color="Maroon" title="c4">x</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="#E1:59_1_1"><span class="lab"><font color="Green" title="E30">A</font></span></a>, <a class="txt" href="#E4:59_1_1"><span class="lab"><font color="Green" title="E33">D</font></span></a>, <a class="ref" href="dilworth.html#D9" target="_self" title="DILWORTH:def.9">Lmin</a></span>;<br/>
<a NAME="E9:59_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#NR4" title="ORDERS_2:NR.4">&gt;</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_1"><span class="lab"><font color="Green" title="E31">B</font></span></a>, <a class="ref" href="waybel_4.html#T57" title="WAYBEL_4:th.57">WAYBEL_4:57</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:59_1_1"/>
not <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E32">C</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</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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