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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon" title="c2">AR</font> be  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>set </b><font color="Maroon" title="c3">c</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</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"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c4">d</font> = <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> ;<br/>
<a NAME="E1:60"/>
 ex <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</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"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></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="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">g</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</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"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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 <span class="p1">(<span class="default"><a href="waybel_4.html#K8" title="WAYBEL_4:func.8">MonSet</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a>  <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 <span class="p1">(<span class="default"><a href="waybel_4.html#K8" title="WAYBEL_4:func.8">MonSet</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Olive" title="b1">f</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Olive" title="b2">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c5">c</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</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"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel_4.html#T25" target="_self" title="WAYBEL_4:th.25">Th25</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">c</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> 
;<br/>

<a NAME="E2:60_1"/>
<font color="Maroon" title="c5">c</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_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="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a><br/>

<b>assume </b><a NAME="E1:60_1_1"/>
<font color="Maroon" title="c6">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="c1">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:60_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c7">x'</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D13" target="_self" title="WAYBEL_4:def.13">Def13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">dx</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x'</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">dx'</font> = <font color="Maroon" title="c8">dx</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:60_1_1"/>
<font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x'</font> <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"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E7:60_1_1"/><b>then </b><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">dx</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E42"><i><font color="Green" title="E30">Lm4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c9">dx'</font>
;<br/>

<b>thus </b><a NAME="E8:60_1_1"/>
( <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c9">dx'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">dx'</font> )
 <b>by </b><i><a class="txt" href="waybel_4.html#E7:60_1_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</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"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></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="c5">c</font> &amp; <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a>  &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</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"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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 <span class="p1">(<span class="default"><a href="waybel_4.html#K8" title="WAYBEL_4:func.8">MonSet</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a>  <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 <span class="p1">(<span class="default"><a href="waybel_4.html#K8" title="WAYBEL_4:func.8">MonSet</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Maroon" title="c5">c</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a>  )
 <b>by </b><i><a class="ref" href="waybel_4.html#T18" target="_self" title="WAYBEL_4:th.18">Th18</a>, <a class="ref" href="waybel_4.html#T26" target="_self" title="WAYBEL_4:th.26">Th26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:60"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" title="WAYBEL_4:func.8">MonSet</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_4.html#D14" target="_self" title="WAYBEL_4:def.14">Def14</a></i>;<br/>


</div>
