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

<b>let </b><font color="Maroon" title="c3">LR1</font>, <font color="Maroon" title="c4">LR2</font> be  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">LR1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">LR1</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> iff  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c1">L</font> ) ) &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">LR2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">LR2</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> iff  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c1">L</font> ) ) implies <font color="Maroon" title="c3">LR1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">LR2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_2"/><i><font color="Green" title="E32">A10</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">LR1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>
 <b>and </b><br/><a NAME="E2:46_1_2"/><i><font color="Green" title="E33">A11</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">LR1</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> iff  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c1">L</font> )
 <b>and </b><br/><a NAME="E3:46_1_2"/><i><font color="Green" title="E34">A12</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">LR2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>
 <b>and </b><br/><a NAME="E4:46_1_2"/><i><font color="Green" title="E35">A13</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">LR2</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> iff  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c1">L</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">LR1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">LR2</font></span><br/>

<b>set </b><font color="Maroon" title="c5">cLR1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">LR1</font>;<br/>
<b>set </b><font color="Maroon" title="c6">cLR2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">LR2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">LR1</font> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">LR2</font> ) &amp; ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">LR2</font> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">LR1</font> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">LR2</font> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">LR1</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:46_1_2_1_1"/><i><font color="Green" title="E36">A14</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">LR1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">LR2</font></span><br/><b>then consider </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:46_1_2_1_1"/><i><font color="Green" title="E37">A15</font></i>: 
( <font color="Maroon" title="c8">x</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="c3">LR1</font> &amp; <font color="Maroon" title="c9">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="c3">LR1</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">LR1</font> <b>by </b><i><a class="txt" href="#E3:46_1_2_1_1"><i><font color="Green" title="E37">A15</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">x'</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y'</font> = <font color="Maroon" title="c11">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">LR2</font> <b>by </b><i><a class="txt" href="#E1:46_1_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="#E3:46_1_2"><i><font color="Green" title="E34">A12</font></i></a></i>;<br/><a NAME="E6:46_1_2_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="#E1:46_1_2_1_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="#E3:46_1_2_1_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E7:46_1_2_1_1"/><b>then </b>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E2:46_1_2"><i><font color="Green" title="E33">A11</font></i></a></i>;<br/><a NAME="E8:46_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c12">x'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">y'</font>
 <b>by </b><i><a class="txt" href="#E4:46_1_2"><i><font color="Green" title="E35">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E9:46_1_2_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">LR2</font>
 <b>by </b><i><a class="txt" href="#E3:46_1_2_1_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:46_1_2_1"/><i><font color="Green" title="E36">A16</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">LR2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">LR1</font></span><br/><b>then consider </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:46_1_2_1"/><i><font color="Green" title="E37">A17</font></i>: 
( <font color="Maroon" title="c8">x</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="c4">LR2</font> &amp; <font color="Maroon" title="c9">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="c4">LR2</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">LR2</font> <b>by </b><i><a class="txt" href="#E4:46_1_2_1"><i><font color="Green" title="E37">A17</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">x'</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y'</font> = <font color="Maroon" title="c11">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">LR1</font> <b>by </b><i><a class="txt" href="#E1:46_1_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="#E3:46_1_2"><i><font color="Green" title="E34">A12</font></i></a></i>;<br/><a NAME="E7:46_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="#E2:46_1_2_1"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="#E4:46_1_2_1"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E8:46_1_2_1"/><b>then </b>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E4:46_1_2"><i><font color="Green" title="E35">A13</font></i></a></i>;<br/><a NAME="E9:46_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">x'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">y'</font>
 <b>by </b><i><a class="txt" href="#E2:46_1_2"><i><font color="Green" title="E33">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E10:46_1_2_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">LR1</font>
 <b>by </b><i><a class="txt" href="#E4:46_1_2_1"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:46_1_2"/>
<font color="Maroon" title="c3">LR1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">LR2</font>
 <b>by </b><i><a class="txt" href="#E1:46_1_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="#E3:46_1_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
