<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c2">LS</font> = <span class="p1">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c3">R</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:71"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:71"/><b>then </b><i><font color="Green" title="E41">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E3:71"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>;<br/>
<a NAME="E4:71"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:71"/><b>then </b><i><font color="Green" title="E42">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E6:71"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<b>assume </b><a NAME="E1:71_1"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">AR</font> = <font color="Maroon" title="c4">a</font> as  <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> <b>by </b><i><a class="ref" href="waybel_4.html#D9" target="_self" title="WAYBEL_4:def.9">Def9</a></i>;<br/>
<a NAME="E3:71_1"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c4">a</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 <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_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E1:71_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E4:71_1"/><b>then </b><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:71_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">AR</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E6:71_1"/><b>then </b><i><font color="Green" title="E46">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D15" target="_self" title="WAYBEL_4:def.15">Def15</a></i>;<br/>
<a NAME="E7:71_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</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 <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_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E3:71_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E8:71_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">La</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> as  <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> <b>by </b><i><a class="ref" href="waybel_4.html#D9" target="_self" title="WAYBEL_4:def.9">Def9</a></i>;<br/>
<a NAME="E10:71_1"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c5">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>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T18" target="_self" title="WAYBEL_4:th.18">Th18</a></i>;<br/>
<a NAME="E11:71_1"/>
<font color="Maroon" title="c6">La</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">AR</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:71_1_1"/>
for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">La</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AR</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">La</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AR</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AR</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">La</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_1_1"/><i><font color="Green" title="E48">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">La</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AR</font></span><br/><b>consider </b><font color="Maroon" title="c9">AR'</font> being  <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><b> such that </b><br/><a NAME="E3:71_1_1_1_1"/><i><font color="Green" title="E49">A9</font></i>: 
( <font color="Maroon" title="c9">AR'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> </span>)</span> &amp; ( for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">AR'</font> iff  ex <font color="Olive" title="b3">c'</font>, <font color="Olive" title="b4">b'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ex <font color="Olive" title="b5">s'</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/>( <font color="Olive" title="b3">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b4">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b5">s'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a>  &amp; <font color="Olive" title="b3">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">s'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">b'</font> ) ) ) )
 <b>by </b><i><a class="txt" href="waybel_4.html#E10:71_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="waybel_4.html#D16" target="_self" title="WAYBEL_4:def.16">Def16</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">c'</font>, <font color="Maroon" title="c11">b'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c12">s'</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><b> such that </b><br/><a NAME="E5:71_1_1_1_1"/><i><font color="Green" title="E50">A10</font></i>: 
( <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c11">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c12">s'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a>  &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">s'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b'</font> )
 <b>by </b><i><a class="txt" href="waybel_4.html#E6:71_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="txt" href="waybel_4.html#E1:71_1_1_1_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="txt" href="waybel_4.html#E3:71_1_1_1_1"><i><font color="Green" title="E49">A9</font></i></a></i>;<br/><a NAME="E6:71_1_1_1_1"/>
<font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c11">b'</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E5:71_1_1_1_1"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="waybel_4.html#D13" target="_self" title="WAYBEL_4:def.13">Def13</a></i>;<br/><b>hence </b><a NAME="E7:71_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AR</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E5:71_1_1_1_1"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="waybel_4.html#T13" target="_self" title="WAYBEL_4:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:71_1_1_1"/><i><font color="Green" title="E48">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AR</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">La</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c9">c'</font> = <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:71_1_1_1"/>
<font color="Maroon" title="c9">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E2:71_1_1_1"><i><font color="Green" title="E48">A11</font></i></a></i>;<br/>
<a NAME="E5:71_1_1_1"/><b>then </b><i><font color="Green" title="E49">A12</font></i>: 
<font color="Maroon" title="c9">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c10">b'</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>consider </b><font color="Maroon" title="c11">AR'</font> being  <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><b> such that </b><br/><a NAME="E7:71_1_1_1"/><i><font color="Green" title="E50">A13</font></i>: 
( <font color="Maroon" title="c11">AR'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> </span>)</span> &amp; ( for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">AR'</font> iff  ex <font color="Olive" title="b3">c'</font>, <font color="Olive" title="b4">b'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ex <font color="Olive" title="b5">s'</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/>( <font color="Olive" title="b3">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b4">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b5">s'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">AR</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a>  &amp; <font color="Olive" title="b3">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">s'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">b'</font> ) ) ) )
 <b>by </b><i><a class="txt" href="waybel_4.html#E10:71_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="waybel_4.html#D16" target="_self" title="WAYBEL_4:def.16">Def16</a></i>;<br/>
<b>thus </b><a NAME="E8:71_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">La</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E6:71_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="txt" href="waybel_4.html#E5:71_1_1_1"><i><font color="Green" title="E49">A12</font></i></a>, <a class="txt" href="waybel_4.html#E7:71_1_1_1"><i><font color="Green" title="E50">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:71_1_1"/>
<font color="Maroon" title="c6">La</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">AR</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:71_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E4:71_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:71"/>
<span class="p1">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_4.html#E2:71"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="waybel_4.html#E5:71"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
