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

<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">T</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 ;<br/><b>let </b><font color="Maroon" title="c3">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>;<br/><a NAME="E2:18_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c3">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> $1,<font color="Maroon" title="c1">T</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="c2">S</font> );<br/><a NAME="E3:18_1"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 ;<br/><div><i><font color="Green" title="E14">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green" title="E15">A5</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>] )
 ;<br/><b>thus </b><a NAME="E2:18_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel21.html#E1:18_1_1"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">z</font> = <font color="Maroon" title="c6">y'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/>
<a NAME="E4:18_1_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:18_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c5">x</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="waybel21.html#E1:18_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:18_1_1_1"/><i><font color="Green" title="E16">A6</font></i>: 
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">z</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c1">T</font> &amp;  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">z</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y'</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T38" title="YELLOW_0:th.38">YELLOW_0:38</a>, <a class="ref" href="yellow_0.html#T39" title="YELLOW_0:th.39">YELLOW_0:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_1_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:18_1_1_1_1"/><b>then </b>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c8">x'</font>,<font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T55" title="YELLOW_0:th.55">YELLOW_0:55</a></i>;<br/><a NAME="E3:18_1_1_1_1"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
(  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <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="c2">S</font> &amp;  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">z</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c8">x'</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c7">z</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<font color="Maroon" title="c7">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E1:18_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="waybel21.html#E6:18_1_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="waybel21.html#E1:18_1_1_1_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="yellow_0.html#T21" title="YELLOW_0:th.21">YELLOW_0:21</a>, <a class="ref" href="yellow_2.html#T4" title="YELLOW_2:th.4">YELLOW_2:4</a></i>;<br/><a NAME="E4:18_1_1_1_1"/><b>then </b>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<font color="Maroon" title="c7">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <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="c2">S</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E1:18_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="yellow_0.html#D16" title="YELLOW_0:def.16">YELLOW_0:def 16</a></i>;<br/><b>hence </b><a NAME="E5:18_1_1_1_1"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">z</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <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="c2">S</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E3:18_1_1_1_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_1_1_1"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c1">T</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="c2">S</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E6:18_1_1_1"><i><font color="Green" title="E16">A6</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:18_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">X</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="waybel21.html#E2:18_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="waybel21.html#E3:18_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="waybel21.html#E4:18_1"><i><font color="Green" title="E14">A4</font></i></a>);<br/></i><b>hence </b><a NAME="E6:18_1"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">T</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="c2">S</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:18"/><i><font color="Green" title="E11">A9</font></i>: 
for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> holds   <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">T</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="c2">S</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D16" title="YELLOW_0:def.16">YELLOW_0:def 16</a><br/>

<b>assume </b><a NAME="E3:18"/>
( <font color="Maroon" title="c3">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="c2">S</font> &amp; <font color="Maroon" title="c4">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="c2">S</font> )
 ;<br/>

<a NAME="E4:18"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/>
<b>hence </b><a NAME="E5:18"/>
(  not  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">T</font> or  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">T</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="c2">S</font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E2:18"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/>


</div>
