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

<b>consider </b><font color="Maroon" title="c2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>
<b>consider </b><font color="Maroon" title="c3">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E3:6_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">e1</font> =  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c5">e2</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">A</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="lattice5.html#T4" title="LATTICE5:th.4">LATTICE5:4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">Y</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>;<br/>
<a NAME="E5:6_1_1"/><i><font color="Green" title="E6">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E6:6_1_1"/>
<font color="Maroon" title="c4">e1</font> <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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D1" title="EQREL_1:def.1">EQREL_1:def 1</a></i>;<br/>
<a NAME="E7:6_1_1"/><b>then </b><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c5">e2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">e1</font>
 
<b>by </b><i><a class="ref" href="lattice5.html#T6" title="LATTICE5:th.6">LATTICE5:6</a></i>;<br/>
<a NAME="E8:6_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:6_1_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font> holds <br/> <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D16" title="YELLOW_0:def.16">YELLOW_0:def 16</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:6_1_1_1_1"/><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_1_1_1"/>
( ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> ) or ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> ) or ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> ) or ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_1_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> = 
<font color="Maroon" title="c4">e1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">e1</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon" title="c4">e1</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>
;<br/><b>hence </b><a NAME="E3:6_1_1_1_1_1_1"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> = 
<font color="Maroon" title="c4">e1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">e2</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon" title="c5">e2</font>
<b>by </b><i><a class="txt" href="lattice8.html#E7:6_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="yellow_5.html#T10" title="YELLOW_5:th.10">YELLOW_5:10</a></i>
;<br/><b>hence </b><a NAME="E3:6_1_1_1_1_1_2"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:6_1_1_1_1_1_3"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> = 
<font color="Maroon" title="c5">e2</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">e1</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon" title="c5">e2</font>
<b>by </b><i><a class="txt" href="lattice8.html#E7:6_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="yellow_5.html#T10" title="YELLOW_5:th.10">YELLOW_5:10</a></i>
;<br/><b>hence </b><a NAME="E3:6_1_1_1_1_1_3"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:6_1_1_1_1_1_4"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> = 
<font color="Maroon" title="c5">e2</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">e2</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon" title="c5">e2</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>
;<br/><b>hence </b><a NAME="E3:6_1_1_1_1_1_4"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:6_1_1_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:6_1_1"/><i><font color="Green" title="E9">A6</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="yellow_0.html#V6" title="YELLOW_0:attr.6">join-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:6_1_1_2"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font> holds <br/> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D17" title="YELLOW_0:def.17">YELLOW_0:def 17</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:6_1_1_2_1"/><i><font color="Green" title="E10">A7</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_2_1_1"/>
( ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> ) or ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> ) or ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> ) or ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_2_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_2_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> = 
<font color="Maroon" title="c4">e1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">e1</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T41" title="YELLOW_0:th.41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon" title="c4">e1</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E3:6_1_1_2_1_1_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_2_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> = 
<font color="Maroon" title="c4">e1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">e2</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T41" title="YELLOW_0:th.41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon" title="c4">e1</font>
<b>by </b><i><a class="txt" href="lattice8.html#E7:6_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="yellow_5.html#T8" title="YELLOW_5:th.8">YELLOW_5:8</a></i>
;<br/><b>hence </b><a NAME="E3:6_1_1_2_1_1_2"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:6_1_1_2_1_1_3"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> = 
<font color="Maroon" title="c5">e2</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">e1</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T41" title="YELLOW_0:th.41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon" title="c4">e1</font>
<b>by </b><i><a class="txt" href="lattice8.html#E7:6_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="yellow_5.html#T8" title="YELLOW_5:th.8">YELLOW_5:8</a></i>
;<br/><b>hence </b><a NAME="E3:6_1_1_2_1_1_3"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:6_1_1_2_1_1_4"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> = 
<font color="Maroon" title="c5">e2</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">e2</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T41" title="YELLOW_0:th.41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon" title="c5">e2</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E3:6_1_1_2_1_1_4"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:6_1_1_2"/>
verum
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:6_1_1"/><i><font color="Green" title="E10">A8</font></i>: 
not  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_1_3"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span><b> such that </b><br/><a NAME="E3:6_1_1_3"/><i><font color="Green" title="E11">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_1.html#D1" title="TEX_1:def.1">TEX_1:def 1</a></i>;<br/>
<a NAME="E4:6_1_1_3"/>
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">A</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="lattice8.html#E3:6_1_1_3"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T9" title="ZFMISC_1:th.9">ZFMISC_1:9</a></i>;<br/>
<a NAME="E5:6_1_1_3"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D1" title="EQREL_1:def.1">EQREL_1:def 1</a></i>;<br/>
<a NAME="E6:6_1_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:6_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice8.html#E3:6_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:6_1_1"/><i><font color="Green" title="E11">A10</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="lattice8.html#V1" title="LATTICE8:attr.1">finitely_typed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_4"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c1">A</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice8.html#D2" target="_self" title="LATTICE8:def.2">LATTICE8:def 2</a><br/>

<b>thus </b><a NAME="E1:6_1_1_4"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> holds <br/><font color="Olive" title="b1">e</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>

<b>take </b><font color="Maroon" title="c7">o</font> = 3;<br/>

<b>thus </b><a NAME="E2:6_1_1_4"/>
for <font color="Olive" title="b1">eq1</font>, <font color="Olive" title="b2">eq2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">y1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">eq1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp; <font color="Olive" title="b2">eq2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x1</font>,<font color="Olive" title="b4">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">eq1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b2">eq2</font> holds <br/> ex <font color="Olive" title="b5">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">A</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o</font> &amp; <font color="Olive" title="b3">x1</font>,<font color="Olive" title="b4">y1</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b5">F</font>,<font color="Olive" title="b1">eq1</font>,<font color="Olive" title="b2">eq2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">eq1</font>, <font color="Maroon" title="c9">eq2</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">y1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:6_1_1_4_1"/><i><font color="Green" title="E12">A11</font></i>: 
( <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">eq1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c9">eq2</font> )
 ;<br/>

<a NAME="E2:6_1_1_4_1"/>
( <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> or <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e2</font> )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c12">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6_1_1_4_1"/><i><font color="Green" title="E13">A12</font></i>: 
( ( <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1</font> ) or ( <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y1</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c13">x2</font>, <font color="Maroon" title="c14">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:6_1_1_4_1"/><i><font color="Green" title="E14">A13</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">y1</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="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x2</font>,<font color="Maroon" title="c14">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:6_1_1_4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/>
<a NAME="E7:6_1_1_4_1"/>
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="lattice8.html#E6:6_1_1_4_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">F</font> = <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c12">z</font>,<font color="Maroon" title="c11">y1</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="lattice8.html#E4:6_1_1_4_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="finseq_2.html#T16" title="FINSEQ_2:th.16">FINSEQ_2:16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">F</font>
;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_4_1_1"/>
( ( <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1</font> ) or ( <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y1</font> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E4:6_1_1_4_1"><i><font color="Green" title="E13">A12</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_4_1_1_1"/><i><font color="Green" title="E15">A14</font></i>: 
( <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1</font> )
 ;<br/><div class="add"><a NAME="E2:6_1_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E16">A15</font></i>: 
( <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1</font> &amp; <font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1</font> &amp; <font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/><a NAME="E3:6_1_1_4_1_1_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">F</font> holds <br/>( ( not <font color="Olive" title="b1">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c8">eq1</font> ) &amp; ( <font color="Olive" title="b1">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c9">eq2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_4_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_1_4_1_1_1_1"/><i><font color="Green" title="E17">A16</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">F</font> )
 ;<br/>

<a NAME="E2:6_1_1_4_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E3:6_1_1_4_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:6_1_1_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E18">A17</font></i>: 
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_4_1_1_1_1_1"/>
( ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> ) or ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> ) or ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> ) or ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_1_1"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="lattice8.html#E4:6_1_1_4_1_1_1_1"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="lattice8.html#E4"><i><font color="Green" title="E3">Lm1</font></i></a>, <a class="txt" href="lattice8.html#E5"><i><font color="Green" title="E4">Lm2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_4_1_1_1_1_1_1"/><i><font color="Green" title="E19">A18</font></i>: 
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c17">x2</font>, <font color="Maroon" title="c18">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1_4_1_1_1_1_1_1"/><i><font color="Green" title="E20">A19</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">y1</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="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">x2</font>,<font color="Maroon" title="c18">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c18">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:6_1_1_4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/><a NAME="E4:6_1_1_4_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:6_1_1_4_1_1_1_1_1_1"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:6_1_1_4_1_1_1_1_1_1"/>
( ( not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c8">eq1</font> ) &amp; ( <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c9">eq2</font> ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E2:6_1_1_4_1_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_1_1_1_1"><i><font color="Green" title="E19">A18</font></i></a>, <a class="ref" href="eqrel_1.html#T11" title="EQREL_1:th.11">EQREL_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_4_1_1_1_1_1_2"/><i><font color="Green" title="E19">A20</font></i>: 
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c17">x2</font>, <font color="Maroon" title="c18">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1_4_1_1_1_1_1_2"/><i><font color="Green" title="E20">A21</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">y1</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="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">x2</font>,<font color="Maroon" title="c18">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c18">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:6_1_1_4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/><a NAME="E4:6_1_1_4_1_1_1_1_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:6_1_1_4_1_1_1_1_1_2"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:6_1_1_4_1_1_1_1_1_2"/>
( ( not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c8">eq1</font> ) &amp; ( <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c9">eq2</font> ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E2:6_1_1_4_1_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_1_1_1_2"><i><font color="Green" title="E19">A20</font></i></a>, <a class="ref" href="eqrel_1.html#T11" title="EQREL_1:th.11">EQREL_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:6_1_1_4_1_1_1_1_1_3"/><i><font color="Green" title="E19">A22</font></i>: 
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> )
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c8">eq1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c9">eq2</font> = 
<font color="Maroon" title="c5">e2</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">e1</font>
<b>by </b><i><a class="ref" href="lattice5.html#T10" title="LATTICE5:th.10">LATTICE5:10</a></i>
<br/>.= 
<font color="Maroon" title="c9">eq2</font>
<b>by </b><i><a class="txt" href="lattice8.html#E7:6_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_1_1_1_3"><i><font color="Green" title="E19">A22</font></i></a>, <a class="ref" href="yellow_5.html#T8" title="YELLOW_5:th.8">YELLOW_5:8</a></i>
;<br/><b>hence </b><a NAME="E3:6_1_1_4_1_1_1_1_1_3"/>
( ( not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c8">eq1</font> ) &amp; ( <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c9">eq2</font> ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:6_1_1_4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="lattice8.html#E2:6_1_1_4_1_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_1_1_1_3"><i><font color="Green" title="E19">A22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:6_1_1_4_1_1_1_1_1_4"/>
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_4_1_1_1_1_1_4"/>
( ( not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c8">eq1</font> ) &amp; ( <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c9">eq2</font> ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:6_1_1_4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="lattice8.html#E2:6_1_1_4_1_1_1"><i><font color="Green" title="E16">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:6_1_1_4_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o</font> &amp; <font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">y1</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Maroon" title="c15">F</font>,<font color="Maroon" title="c8">eq1</font>,<font color="Maroon" title="c9">eq2</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E2:6_1_1_4_1_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_4_1_1_2"/><i><font color="Green" title="E15">A23</font></i>: 
( <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e2</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y1</font> )
 ;<br/><div class="add"><a NAME="E2:6_1_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E16">A24</font></i>: 
( <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1</font> &amp; <font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y1</font> &amp; <font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/><a NAME="E3:6_1_1_4_1_1_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">F</font> holds <br/>( ( not <font color="Olive" title="b1">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c8">eq1</font> ) &amp; ( <font color="Olive" title="b1">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c9">eq2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_4_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_1_4_1_1_2_1"/><i><font color="Green" title="E17">A25</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">F</font> )
 ;<br/>

<a NAME="E2:6_1_1_4_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E3:6_1_1_4_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:6_1_1_4_1_1_2_1"/><b>then </b><i><font color="Green" title="E18">A26</font></i>: 
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_4_1_1_2_1_1"/>
( ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> ) or ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> ) or ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> ) or ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E5:6_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_2"><i><font color="Green" title="E15">A23</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_2_1"><i><font color="Green" title="E17">A25</font></i></a>, <a class="txt" href="lattice8.html#E4:6_1_1_4_1_1_2_1"><i><font color="Green" title="E18">A26</font></i></a>, <a class="txt" href="lattice8.html#E4"><i><font color="Green" title="E3">Lm1</font></i></a>, <a class="txt" href="lattice8.html#E5"><i><font color="Green" title="E4">Lm2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_4_1_1_2_1_1_1"/>
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_4_1_1_2_1_1_1"/>
( ( not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c8">eq1</font> ) &amp; ( <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c9">eq2</font> ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:6_1_1_4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="lattice8.html#E2:6_1_1_4_1_1_2"><i><font color="Green" title="E16">A24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_4_1_1_2_1_1_2"/><i><font color="Green" title="E19">A27</font></i>: 
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> )
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c8">eq1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c9">eq2</font> = 
<font color="Maroon" title="c4">e1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">e2</font>
<b>by </b><i><a class="ref" href="lattice5.html#T10" title="LATTICE5:th.10">LATTICE5:10</a></i>
<br/>.= 
<font color="Maroon" title="c8">eq1</font>
<b>by </b><i><a class="txt" href="lattice8.html#E7:6_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_2_1_1_2"><i><font color="Green" title="E19">A27</font></i></a>, <a class="ref" href="yellow_5.html#T8" title="YELLOW_5:th.8">YELLOW_5:8</a></i>
;<br/><b>hence </b><a NAME="E3:6_1_1_4_1_1_2_1_1_2"/>
( ( not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c8">eq1</font> ) &amp; ( <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c9">eq2</font> ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:6_1_1_4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="lattice8.html#E2:6_1_1_4_1_1_2"><i><font color="Green" title="E16">A24</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_2_1_1_2"><i><font color="Green" title="E19">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:6_1_1_4_1_1_2_1_1_3"/><i><font color="Green" title="E19">A28</font></i>: 
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c17">x2</font>, <font color="Maroon" title="c18">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1_4_1_1_2_1_1_3"/><i><font color="Green" title="E20">A29</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">y1</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="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">x2</font>,<font color="Maroon" title="c18">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c18">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:6_1_1_4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/><a NAME="E4:6_1_1_4_1_1_2_1_1_3"/>
<font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:6_1_1_4_1_1_2_1_1_3"><i><font color="Green" title="E20">A29</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:6_1_1_4_1_1_2_1_1_3"/>
( ( not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c8">eq1</font> ) &amp; ( <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c9">eq2</font> ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E2:6_1_1_4_1_1_2"><i><font color="Green" title="E16">A24</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_2_1_1_3"><i><font color="Green" title="E19">A28</font></i></a>, <a class="ref" href="eqrel_1.html#T11" title="EQREL_1:th.11">EQREL_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:6_1_1_4_1_1_2_1_1_4"/><i><font color="Green" title="E19">A30</font></i>: 
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c8">eq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e1</font> &amp; <font color="Maroon" title="c9">eq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c17">x2</font>, <font color="Maroon" title="c18">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1_4_1_1_2_1_1_4"/><i><font color="Green" title="E20">A31</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">y1</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="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">x2</font>,<font color="Maroon" title="c18">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c18">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:6_1_1_4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/><a NAME="E4:6_1_1_4_1_1_2_1_1_4"/>
<font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:6_1_1_4_1_1_2_1_1_4"><i><font color="Green" title="E20">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:6_1_1_4_1_1_2_1_1_4"/>
( ( not <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c8">eq1</font> ) &amp; ( <font color="Maroon" title="c16">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <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="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c9">eq2</font> ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E2:6_1_1_4_1_1_2"><i><font color="Green" title="E16">A24</font></i></a>, <a class="txt" href="lattice8.html#E1:6_1_1_4_1_1_2_1_1_4"><i><font color="Green" title="E19">A30</font></i></a>, <a class="ref" href="eqrel_1.html#T11" title="EQREL_1:th.11">EQREL_1:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:6_1_1_4_1_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o</font> &amp; <font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">y1</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Maroon" title="c15">F</font>,<font color="Maroon" title="c8">eq1</font>,<font color="Maroon" title="c9">eq2</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E2:6_1_1_4_1_1_2"><i><font color="Green" title="E16">A24</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c7">Y</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of  <a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="lattice8.html#E8:6_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="lattice8.html#E9:6_1_1"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">Y</font>
;<br/>

<b>thus </b><a NAME="E13:6_1_1"/>
( not <font color="Maroon" title="c7">Y</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Maroon" title="c7">Y</font> is <a href="lattice8.html#V1" title="LATTICE8:attr.1">finitely_typed</a> &amp; <font color="Maroon" title="c7">Y</font> is <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> )
 <b>by </b><i><a class="txt" href="lattice8.html#E10:6_1_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="lattice8.html#E11:6_1_1"><i><font color="Green" title="E11">A10</font></i></a></i>;<br/>


</div>
