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

<a NAME="E1:22_1_1"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c1">r</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">Relation</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T8" title="RELSET_1:th.8">RELSET_1:8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">T</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">r</font> #) as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">T</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">T</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  &amp; not <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E4:22_1_1"/>
<font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</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="c2">T</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  &amp; not <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E5:22_1_1"/>
<font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</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="c2">T</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  &amp; not <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D1" title="YELLOW_0:def.1">YELLOW_0:def 1</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="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font></span><br/>

<a NAME="E1:22_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E2:22_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:22_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</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>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">T9</font> = <font color="Maroon" title="c2">T</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">T99</font> = <font color="Maroon" title="c3">T9</font> as   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ;<br/>
<a NAME="E8:22_1_1"/>
<font color="Maroon" title="c3">T9</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E9:22_1_1"/>
<font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</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="c2">T</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  &amp; not <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><br/>

<a NAME="E10:22_1_1"/>
<font color="Maroon" title="c3">T9</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E11:22_1_1"/>
<font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</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="c2">T</font> is  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  &amp; not <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><br/>

<a NAME="E12:22_1_1"/>
<font color="Maroon" title="c3">T9</font> is  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E13:22_1_1"/>
<font color="Maroon" title="c2">T</font> is  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</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="c2">T</font> is  <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  &amp; not <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><br/>

<a NAME="E14:22_1_1"/>
<font color="Maroon" title="c3">T9</font> is  <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E15:22_1_1"/>
<font color="Maroon" title="c2">T</font> is  <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</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="c2">T</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  &amp; not <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><br/>

<a NAME="E16:22_1_1"/>
<font color="Maroon" title="c4">T99</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E17:22_1_1"/>
<font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> 
 ; <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="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E18:22_1_1"/>
not <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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="c2">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  &amp; <font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E19:22_1_1"/>
<font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</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="c2">T</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> </span><br/>

<span class="kw">thus </span><a NAME="E20:22_1_1"/>
<font color="Maroon" title="c2">T</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> 
 ; <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>
