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

<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V2">with_infima</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>



<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>set </b><font color="Maroon">N</font> =  <a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<a NAME="E2:22"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</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">j</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i1</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> )
 ;<br/>
<a NAME="E4:22_1_1"/>
( <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:22_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">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">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">J</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Olive">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">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">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">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">J</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">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">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">J</font>]
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="waybel17.html#K1" target="_self">,...</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, </i>
;<br/><a NAME="E3:22_1_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">j</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:22_1_2_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">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">E52</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">J</font>]
 ;<br/><div class="add"><i><font color="Green">E53</font></i>: <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="waybel17.html#K1" target="_self">,...</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, </i>
;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1 as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E4:22_1_2_1_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:22_1_2_1_1_1_1_2"/>
<font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:22_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:22_1_2_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 <b>by </b><i>, </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">x</font> <a href="hidden.html#R2">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">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">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">J</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">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">E58</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">J</font>]
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="waybel17.html#K1" target="_self">,...</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, </i>
;<br/><a NAME="E3:22_1_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">j</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:22_1_2_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">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">E105</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">J</font>]
 ;<br/><div class="add"><i><font color="Green">E110</font></i>: <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="waybel17.html#K1" target="_self">,...</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, </i>
;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1 as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E4:22_1_2_1_2_1_1_2"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:22_1_2_1_2_1_1_2"/>
<font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:22_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:22_1_2_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 <b>by </b><i>, </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">x</font> <a href="hidden.html#R2">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">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">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i1</font></span>)</span> where <font color="Olive">i1</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i1</font>,<font color="Maroon">a<sub>1</sub></font>] </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> =  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>),<font color="Maroon">S</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> =  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a<sub>1</sub></font>),<font color="Maroon">S</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> = <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>;<br/>
<a NAME="E3:22"/><i><font color="Green">E114</font></i>: 
for <font color="Olive">jj</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">jj</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">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">jj</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>;<br/>

<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">jj</font>) = 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">jj</font>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">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">jj</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">jj</font>)
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:22"/><i><font color="Green">E116</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">j3</font>) where <font color="Olive">j3</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j3</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">j4</font>) where <font color="Olive">j4</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j4</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S5">FRAENKEL:sch 5</a>();<br/></i>
<a NAME="E5:22"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> where <font color="Olive">j4</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j4</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K6">}</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">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> where <font color="Olive">j4</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j4</font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> where <font color="Olive">j4</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j4</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span><b> such that </b><br/><a NAME="E3:22_3_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">q</font>] )
 ;<br/>
<b>thus </b><a NAME="E4:22_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">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">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> where <font color="Olive">j4</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

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


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


</div><b>end;</b></div>
<a NAME="E6:22"/><b>then </b>
 <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>
<b>hence </b><a NAME="E7:22"/>
 <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
 ;<br/>


</div>
