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

<span class="kw">set </span><font color="Maroon" title="c2">A</font> =  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E1:22_1_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font> is  <a href="waybel_4.html#V1" title="WAYBEL_4:attr.1">auxiliary(i)</a> 
 
<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>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_4.html#D4" target="_self" title="WAYBEL_4:def.4">WAYBEL_4:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_1_1_1"/>
<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="c4">y</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>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</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_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font></span><br/>

<a NAME="E2:22_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_4.html#D10" target="_self" title="WAYBEL_4:def.10">Def10</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="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</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>
<a NAME="E2:22_1_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font> is  <a href="waybel_4.html#V2" title="WAYBEL_4:attr.2">auxiliary(ii)</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font>, <font color="Maroon" title="c6">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_4.html#D5" target="_self" title="WAYBEL_4:def.5">WAYBEL_4:def 5</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="c6">u</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_1_1_2"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">and </span><br/><a NAME="E2:22_1_1_2"/><span class="lab"><font color="Green" title="E16">A4</font></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="c4">y</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>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E3:22_1_1_2"/>
<font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E4:22_1_1_2"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1_1_2"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="waybel_4.html#D10" target="_self" title="WAYBEL_4:def.10">Def10</a></span>;<br/>
<a NAME="E5:22_1_1_2"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/>
<a NAME="E6:22_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1_2"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:22_1_1_2"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:22_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_4.html#D10" target="_self" title="WAYBEL_4:def.10">Def10</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>
<a NAME="E3:22_1_1"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
 <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font> is  <a href="waybel_4.html#V3" title="WAYBEL_4:attr.3">auxiliary(iii)</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_4.html#D6" target="_self" title="WAYBEL_4:def.6">WAYBEL_4:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="c5">z</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>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</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>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c5">z</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>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_1_1_3"/><span class="lab"><font color="Green" title="E16">A7</font></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="c5">z</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>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E2:22_1_1_3"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</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>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c5">z</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>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E3:22_1_1_3"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1_1_3"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="waybel_4.html#D10" target="_self" title="WAYBEL_4:def.10">Def10</a></span>;<br/>
<a NAME="E4:22_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1_3"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="waybel_4.html#D10" target="_self" title="WAYBEL_4:def.10">Def10</a></span>;<br/>
<a NAME="E5:22_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_1_3"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:22_1_1_3"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c5">z</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>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_4.html#D10" target="_self" title="WAYBEL_4:def.10">Def10</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>
<a NAME="E4:22_1_1"/>
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">L</font> holds  <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">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>  <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_4.html#D10" target="_self" title="WAYBEL_4:def.10">Def10</a></span>;<br/>
<a NAME="E5:22_1_1"/><span class="kw">then </span>
 <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font> is  <a href="waybel_4.html#V4" title="WAYBEL_4:attr.4">auxiliary(iv)</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_4.html#D7" target="_self" title="WAYBEL_4:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:22_1_1"/>
 <a href="waybel_4.html#K4" title="WAYBEL_4:func.4">AuxBottom</a> <font color="Maroon" title="c1">L</font> is  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:22_1_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:22_1_1"><span class="lab"><font color="Green" title="E15">A6</font></span></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>
