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

<span class="kw">let </span><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>,<span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">R</font> =  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S</font> =  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">D1</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">D2</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">p2</font> =  <a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">p1</font> =  <a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>);<br/>
<span class="kw">take </span><font color="Maroon" title="c9">f</font> = <span class="p1"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="wellord1.html#D8" title="WELLORD1:def.8">WELLORD1:def 8</a>,<a class="ref" href="filter_1.html#D9" target="_self" title="FILTER_1:def.9">FILTER_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>, <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span><br/>

<a NAME="E1:68"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a></span>;<br/>
<a NAME="E2:68"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></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"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T32" target="_self" title="FILTER_1:th.32">Th33</a></span>;<br/>
<a NAME="E3:68"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</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 <font color="Maroon" title="c2">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T46" title="FUNCT_3:th.46">FUNCT_3:46</a></span>;<br/>
<a NAME="E4:68"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></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"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T32" target="_self" title="FILTER_1:th.32">Th33</a></span>;<br/>
<a NAME="E5:68"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" title="FUNCT_3:def.4">FUNCT_3:def 4</a></span>;<br/>
<a NAME="E6:68"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E42">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:68"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="funct_3.html#D7" title="FUNCT_3:def.7">FUNCT_3:def 7</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <font color="Maroon" title="c9">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> ) ) &amp; (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> ) ) ) )</span><br/>

<a NAME="E8:68"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</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 <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T44" title="FUNCT_3:th.44">FUNCT_3:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:68"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="funct_3.html#T51" title="FUNCT_3:th.51">FUNCT_3:51</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> ) ) &amp; (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E10:68"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> ) ) &amp; (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> ) ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:68_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>, <font color="Maroon" title="c12">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><span class="kw"> such that </span><br/><a NAME="E3:68_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c11">r2</font>,<font color="Maroon" title="c12">r1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/>
<a NAME="E4:68_1"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">r1</font>,<font color="Maroon" title="c11">r2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a></span>;<br/>
<a NAME="E5:68_1"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">r1</font>,<font color="Maroon" title="c11">r2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" title="FUNCT_3:def.4">FUNCT_3:def 4</a></span>;<br/>
<a NAME="E6:68_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c12">r1</font>,<font color="Maroon" title="c11">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:68_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E3:68_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="txt" href="#E4:68_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="txt" href="#E5:68_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="funct_3.html#D7" title="FUNCT_3:def.7">FUNCT_3:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E11:68"/>
<font color="Maroon" title="c9">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> ) ) &amp; (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">f</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">f</font> or  not <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">f</font> or  not <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">f</font> or  not <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:68_2"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">f</font> or  not <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> )</span><br/>

<a NAME="E2:68_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A10</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" title="FUNCT_3:def.7">FUNCT_3:def 7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c13">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E4:68_2"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c12">r1</font>,<font color="Maroon" title="c13">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E1:68_2"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/>
<a NAME="E5:68_2"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">r1</font>,<font color="Maroon" title="c13">r2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a></span>;<br/>
<a NAME="E6:68_2"/><span class="lab"><font color="Green" title="E51">A13</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">r1</font>,<font color="Maroon" title="c13">r2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" title="FUNCT_3:def.4">FUNCT_3:def 4</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:68_2"/><span class="lab"><font color="Green" title="E52">A14</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">and </span><br/><a NAME="E8:68_2"/><span class="lab"><font color="Green" title="E53">A15</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/>

<a NAME="E9:68_2"/><span class="lab"><font color="Green" title="E54">A16</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68_2"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="funct_3.html#D7" title="FUNCT_3:def.7">FUNCT_3:def 7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c15">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E11:68_2"/><span class="lab"><font color="Green" title="E55">A17</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c14">q1</font>,<font color="Maroon" title="c15">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E7:68_2"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/>
<a NAME="E12:68_2"/><span class="lab"><font color="Green" title="E56">A18</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">q1</font>,<font color="Maroon" title="c15">q2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a></span>;<br/>
<a NAME="E13:68_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">q1</font>,<font color="Maroon" title="c15">q2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" title="FUNCT_3:def.4">FUNCT_3:def 4</a></span>;<br/>
<a NAME="E14:68_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_2"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E8:68_2"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E11:68_2"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E6:68_2"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E2:68_2"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E9:68_2"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:68_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_2"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E8:68_2"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E11:68_2"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E5:68_2"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="txt" href="#E12:68_2"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E2:68_2"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E9:68_2"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> ) ) &amp; (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> ) )</span><br/>



<span class="kw">thus </span><a NAME="E12:68"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> implies ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:68_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span><span class="kw"> such that </span><br/><a NAME="E3:68_3"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:68_3"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">b</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c15">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E6:68_3"/><span class="lab"><font color="Green" title="E49">A21</font></span>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c14">q1</font>,<font color="Maroon" title="c15">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/>
<a NAME="E7:68_3"/><span class="lab"><font color="Green" title="E50">A22</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">q1</font>,<font color="Maroon" title="c15">q2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">q1</font>,<font color="Maroon" title="c15">q2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">q1</font>,<font color="Maroon" title="c15">q2</font>)</span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E6:68_3"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="ref" href="funct_3.html#D7" title="FUNCT_3:def.7">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E8:68_3"/><span class="lab"><font color="Green" title="E51">A23</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">q1</font>,<font color="Maroon" title="c15">q2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c17">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E10:68_3"/><span class="lab"><font color="Green" title="E52">A24</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c16">r1</font>,<font color="Maroon" title="c17">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/>
<a NAME="E11:68_3"/><span class="lab"><font color="Green" title="E53">A25</font></span>: 
<font color="Maroon" title="c17">r2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_3"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E10:68_3"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="txt" href="#E6:68_3"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th37</a></span>;<br/>
<a NAME="E12:68_3"/>
<font color="Maroon" title="c16">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_3"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E10:68_3"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="txt" href="#E6:68_3"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th37</a></span>;<br/>
<a NAME="E13:68_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A26</font></span>: 
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c17">r2</font>,<font color="Maroon" title="c16">r1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c15">q2</font>,<font color="Maroon" title="c14">q1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68_3"><span class="lab"><font color="Green" title="E53">A25</font></span></a>, <a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th37</a></span>;<br/>
<a NAME="E14:68_3"/><span class="lab"><font color="Green" title="E55">A27</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c16">r1</font>,<font color="Maroon" title="c17">r2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" title="FUNCT_3:def.4">FUNCT_3:def 4</a></span>;<br/>
<a NAME="E15:68_3"/><span class="lab"><font color="Green" title="E56">A28</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c16">r1</font>,<font color="Maroon" title="c17">r2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a></span>;<br/>
<a NAME="E16:68_3"/><span class="lab"><font color="Green" title="E57">A29</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_3"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E17:68_3"/><span class="lab"><font color="Green" title="E58">A30</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_3"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:68_3"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E16:68_3"><span class="lab"><font color="Green" title="E57">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span><br/>

<a NAME="E19:68_3"/><span class="lab"><font color="Green" title="E59">A31</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">q1</font>,<font color="Maroon" title="c15">q2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" title="FUNCT_3:def.4">FUNCT_3:def 4</a></span>;<br/>
<a NAME="E20:68_3"/>
<font color="Maroon" title="c9">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c16">r1</font>,<font color="Maroon" title="c17">r2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c16">r1</font>,<font color="Maroon" title="c17">r2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c16">r1</font>,<font color="Maroon" title="c17">r2</font>)</span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E10:68_3"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="ref" href="funct_3.html#D7" title="FUNCT_3:def.7">FUNCT_3:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:68_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:68_3"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="txt" href="#E6:68_3"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="txt" href="#E17:68_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="txt" href="#E16:68_3"><span class="lab"><font color="Green" title="E57">A29</font></span></a>, <a class="txt" href="#E13:68_3"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E14:68_3"><span class="lab"><font color="Green" title="E55">A27</font></span></a>, <a class="txt" href="#E15:68_3"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="txt" href="#E19:68_3"><span class="lab"><font color="Green" title="E59">A31</font></span></a>, <a class="txt" href="#E8:68_3"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="txt" href="#E7:68_3"><span class="lab"><font color="Green" title="E50">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E13:68"/><span class="lab"><font color="Green" title="E47">A32</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:68"/><span class="lab"><font color="Green" title="E48">A33</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p2"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c13">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E16:68"/><span class="lab"><font color="Green" title="E49">A34</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c13">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E14:68"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/>
<a NAME="E17:68"/><span class="lab"><font color="Green" title="E50">A35</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c13">q2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c13">q2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c13">q2</font>)</span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E16:68"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="ref" href="funct_3.html#D7" title="FUNCT_3:def.7">FUNCT_3:def 7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E18:68"/><span class="lab"><font color="Green" title="E51">A36</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></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>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span></span><br/>

<a NAME="E19:68"/><span class="lab"><font color="Green" title="E52">A37</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c13">q2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a></span>;<br/>
<a NAME="E20:68"/><span class="lab"><font color="Green" title="E53">A38</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c13">q2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" title="FUNCT_3:def.4">FUNCT_3:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c15">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><span class="kw"> such that </span><br/><a NAME="E22:68"/><span class="lab"><font color="Green" title="E54">A39</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c14">r1</font>,<font color="Maroon" title="c15">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E13:68"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/>
<a NAME="E23:68"/><span class="lab"><font color="Green" title="E55">A40</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">r1</font>,<font color="Maroon" title="c15">r2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a></span>;<br/>
<a NAME="E24:68"/><span class="lab"><font color="Green" title="E56">A41</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">r1</font>,<font color="Maroon" title="c15">r2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" title="FUNCT_3:def.4">FUNCT_3:def 4</a></span>;<br/>
<a NAME="E25:68"/>
<font color="Maroon" title="c9">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">r1</font>,<font color="Maroon" title="c15">r2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">r1</font>,<font color="Maroon" title="c15">r2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c14">r1</font>,<font color="Maroon" title="c15">r2</font>)</span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E22:68"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="ref" href="funct_3.html#D7" title="FUNCT_3:def.7">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E26:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A42</font></span>: 
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c15">r2</font>,<font color="Maroon" title="c14">r1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c13">q2</font>,<font color="Maroon" title="c12">q1</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:68"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E16:68"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="txt" href="#E18:68"><span class="lab"><font color="Green" title="E51">A36</font></span></a>, <a class="txt" href="#E24:68"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E23:68"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E17:68"><span class="lab"><font color="Green" title="E50">A35</font></span></a>, <a class="txt" href="#E20:68"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E19:68"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="ref" href="filter_1.html#T31" target="_self" title="FILTER_1:th.31">Th32</a></span>;<br/>
<a NAME="E27:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A43</font></span>: 
<font color="Maroon" title="c14">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th37</a></span>;<br/>
<a NAME="E28:68"/>
<font color="Maroon" title="c15">r2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:68"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th37</a></span>;<br/>
<a NAME="E29:68"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c14">r1</font>,<font color="Maroon" title="c15">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c13">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:68"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:68"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_1.html#K8" title="FILTER_1:func.8">LattRel</a> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:68"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E16:68"><span class="lab"><font color="Green" title="E49">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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