<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">x</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Olive" title="b2">S</font> holds <br/><font color="Olive" title="b1">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><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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"> for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>  st <font color="Maroon" title="c2">x</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Maroon" title="c2">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><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <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"> ( <font color="Maroon" title="c2">x</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c3">S</font> implies <font color="Maroon" title="c2">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><br/>





<span class="kw">assume </span><a NAME="E1:55"/><span class="lab"><font color="Green" title="E27">A</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c3">S</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="c2">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><br/>

<span class="kw">set </span><font color="Maroon" title="c4">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/>
<a NAME="E2:55"/><span class="lab"><font color="Green" title="E28">B</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E27">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">assume </span><a NAME="E3:55"/>
not <font color="Maroon" title="c2">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E5:55"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E6:55"/><span class="lab"><font color="Green" title="E29">E</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E28">B</font></span></a>, <a class="ref" href="waybel_4.html#T57" title="WAYBEL_4:th.57">WAYBEL_4:57</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:55_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c3">S</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E28">B</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1"><span class="kw">suppose </span></a><a NAME="E1:55_1_1"/><span class="lab"><font color="Green" title="E30">S1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:55_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E29">E</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>;<br/><a NAME="E3:55_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E30">S1</font></span></a>, <a class="ref" href="waybel_0.html#D15" title="WAYBEL_0:def.15">WAYBEL_0:def 15</a></span>;<br/><a NAME="E4:55_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:55_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E27">A</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E29">E</font></span></a>, <a class="ref" href="waybel_4.html#T57" title="WAYBEL_4:th.57">WAYBEL_4:57</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2"><span class="kw">suppose </span></a><a NAME="E1:55_1_2"/><span class="lab"><font color="Green" title="E30">S1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">x99</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:55_1_2"/><span class="lab"><font color="Green" title="E31">H</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x99</font>
 <span class="kw">and </span><br/><a NAME="E4:55_1_2"/><span class="lab"><font color="Green" title="E32">I</font></span>: 
<font color="Maroon" title="c6">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2"><span class="lab"><font color="Green" title="E30">S1</font></span></a>, <a class="ref" href="waybel_0.html#D15" title="WAYBEL_0:def.15">WAYBEL_0:def 15</a></span>;<br/><a NAME="E5:55_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E29">E</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>;<br/><a NAME="E6:55_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">J</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55_1_2"><span class="lab"><font color="Green" title="E31">H</font></span></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></span>;<br/><a NAME="E7:55_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_1_2"><span class="lab"><font color="Green" title="E32">I</font></span></a>, <a class="txt" href="#E6:55_1_2"><span class="lab"><font color="Green" title="E33">J</font></span></a>, <a class="ref" href="waybel_0.html#D15" title="WAYBEL_0:def.15">WAYBEL_0:def 15</a></span>;<br/><a NAME="E8:55_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dilworth.html#K3" title="DILWORTH:func.3">Lower</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E9:55_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E27">A</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E29">E</font></span></a>, <a class="ref" href="waybel_4.html#T57" title="WAYBEL_4:th.57">WAYBEL_4:57</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
