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

<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>



<b>set </b><font color="Maroon" title="c4">N</font> =  <a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>
<a NAME="E1:22"/><i><font color="Green" title="E14">A1</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> holds  <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>;<br/>

<b>thus </b><a NAME="E1:22_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">i1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_1"/><i><font color="Green" title="E15">A2</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> )
 ;<br/>
<a NAME="E4:22_1_1"/>
( <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="ref" href="waybel17.html#T5" target="_self" title="WAYBEL17:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E5:22_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="txt" href="waybel17.html#E3:22_1_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:22_1"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1_2"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">J</font> = <font color="Maroon" title="c5">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">k</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_2_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 </span><b>by </b><i><a class="txt" href="waybel17.html#E1:22_1_2"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_1"><b>suppose </b></a><a NAME="E1:22_1_2_1_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_2_1_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">J</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">J</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_2_1_1_1_1_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">J</font>]
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="waybel17.html#K1" title="WAYBEL17:func.1">,...</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="waybel17.html#E1:22_1_2_1_1_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="waybel17.html#D1" target="_self" title="WAYBEL17:def.1">Def1</a></i>
;<br/><a NAME="E3:22_1_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> )
 <b>by </b><i><a class="txt" href="waybel17.html#E1:22_1_2_1_1"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E4:22_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_2_1_1_1_1_2"/><i><font color="Green" title="E17">A6</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">J</font>]
 ;<br/><div class="add"><i><font color="Green" title="E18">A7</font></i>: <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="waybel17.html#K1" title="WAYBEL17:func.1">,...</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="waybel17.html#E1:22_1_2_1_1_1_1_2"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="waybel17.html#D1" target="_self" title="WAYBEL17:def.1">Def1</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c7">J</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/><a NAME="E4:22_1_2_1_1_1_1_2"/><i><font color="Green" title="E19">A8</font></i>: 
<span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E2:22_1_2_1_1_1_1_2"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="waybel17.html#T6" target="_self" title="WAYBEL17:th.6">Th6</a></i>;<br/><a NAME="E5:22_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c7">J</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">J</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:22_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E7:22_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel17.html#E1:22_1_2_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="waybel17.html#E4:22_1_2_1_1_1_1_2"><i><font color="Green" title="E19">A8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:22_1_2_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_2"><b>suppose </b></a><a NAME="E1:22_1_2_1_2"/><i><font color="Green" title="E16">A9</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_2_1_2_1_1"/>
(  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">J</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">J</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_2_1_2_1_1_1"/><i><font color="Green" title="E17">A10</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">J</font>]
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="waybel17.html#K1" title="WAYBEL17:func.1">,...</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="waybel17.html#E1:22_1_2_1_2_1_1_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="waybel17.html#D1" target="_self" title="WAYBEL17:def.1">Def1</a></i>
;<br/><a NAME="E3:22_1_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> )
 <b>by </b><i><a class="txt" href="waybel17.html#E1:22_1_2_1_2"><i><font color="Green" title="E16">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E4:22_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_2_1_2_1_1_2"/><i><font color="Green" title="E17">A11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">J</font>]
 ;<br/><div class="add"><i><font color="Green" title="E18">A12</font></i>: <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="waybel17.html#K1" title="WAYBEL17:func.1">,...</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="waybel17.html#E1:22_1_2_1_2_1_1_2"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="waybel17.html#D1" target="_self" title="WAYBEL17:def.1">Def1</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c7">J</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/><a NAME="E4:22_1_2_1_2_1_1_2"/><i><font color="Green" title="E19">A13</font></i>: 
<span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E2:22_1_2_1_2_1_1_2"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="waybel17.html#T6" target="_self" title="WAYBEL17:th.6">Th6</a></i>;<br/><a NAME="E5:22_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c7">J</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">J</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:22_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E7:22_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel17.html#E1:22_1_2_1_2"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="waybel17.html#E4:22_1_2_1_2_1_1_2"><i><font color="Green" title="E19">A13</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:22_1_2_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> </span>}</span> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>] <b>means </b>$1 <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> $2;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i1</font></span>)</span> where <font color="Olive" title="b1">i1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i1</font>,$1] </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> =  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>($1),<font color="Maroon" title="c1">S</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> =  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>($1),<font color="Maroon" title="c1">S</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> = <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font>;<br/>
<a NAME="E2:22"/><i><font color="Green" title="E15">A14</font></i>: 
for <font color="Olive" title="b1">jj</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">jj</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">jj</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">jj</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>;<br/>

<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c5">jj</font>) = 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c5">jj</font>)
<b>by </b><i><a class="txt" href="waybel17.html#E1:22"><i><font color="Green" title="E14">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></i>
;<br/>
<b>hence </b><a NAME="E2:22_2"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c5">jj</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c5">jj</font>)
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:22"/><i><font color="Green" title="E16">A15</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">j3</font>) where <font color="Olive" title="b1">j3</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j3</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">j4</font>) where <font color="Olive" title="b1">j4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j4</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S5" title="FRAENKEL:sch.5">FRAENKEL:sch 5</a>(<a class="txt" href="waybel17.html#E2:22"><i><font color="Green" title="E15">A14</font></i></a>);<br/></i>
<a NAME="E4:22"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> where <font color="Olive" title="b1">j4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j4</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> where <font color="Olive" title="b1">j4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j4</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> where <font color="Olive" title="b1">j4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j4</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span><b> such that </b><br/><a NAME="E3:22_3_1"/><i><font color="Green" title="E17">A16</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">q</font>] )
 ;<br/>
<b>thus </b><a NAME="E4:22_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="waybel17.html#E3:22_3_1"><i><font color="Green" title="E17">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:22_3"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> where <font color="Olive" title="b1">j4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j4</font>] </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_3_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:22_3_2"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:22_3_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> where <font color="Olive" title="b1">j4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j4</font>] </span>}</span> 
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:22"/>
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E3:22"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="yellow_0.html#T39" title="YELLOW_0:th.39">YELLOW_0:39</a></i>;<br/>


</div>
