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

<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">a</font></span><a href="realset1.html#K5">}</a></span>;<br/>
<a NAME="E3:7_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e1</font> =  <a href="eqrel_1.html#K1">nabla</a> <font color="Maroon">A</font>, <font color="Maroon">e2</font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">A</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="lattice5.html#T4">LATTICE5:4</a></i>;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span>;<br/>
<a NAME="E5:7_1_1"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E6:7_1_1"/>
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D1">EQREL_1:def 1</a></i>;<br/>
<a NAME="E7:7_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">e2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">e1</font>
 
<b>by </b><i><a class="ref" href="lattice5.html#T6">LATTICE5:6</a></i>;<br/>
<a NAME="E8:7_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span> is <a href="yellow_0.html#V5">meet-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

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

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:7_1_1_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>, <a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1_1_1_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> = 
<font color="Maroon">e1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">e1</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon">e1</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>
;<br/><b>hence </b><a NAME="E3:7_1_1_1_1_1_1"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> = 
<font color="Maroon">e1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">e2</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon">e2</font>
<b>by </b><i>, <a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>
;<br/><b>hence </b><a NAME="E3:7_1_1_1_1_1_2"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> = 
<font color="Maroon">e2</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">e1</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon">e2</font>
<b>by </b><i>, <a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>
;<br/><b>hence </b><a NAME="E3:7_1_1_1_1_1_3"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> = 
<font color="Maroon">e2</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">e2</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon">e2</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>
;<br/><b>hence </b><a NAME="E3:7_1_1_1_1_1_4"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">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="E2:7_1_1_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:7_1_1"/><i><font color="Green">E61</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span> is <a href="yellow_0.html#V6">join-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2"><b>proof </b></a><div class="add">

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

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:7_1_1_2_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>, <a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1_1_2_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> = 
<font color="Maroon">e1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">e1</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon">e1</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E3:7_1_1_2_1_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_2_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> = 
<font color="Maroon">e1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">e2</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon">e1</font>
<b>by </b><i>, <a class="ref" href="yellow_5.html#T8">YELLOW_5:8</a></i>
;<br/><b>hence </b><a NAME="E3:7_1_1_2_1_1_2"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:7_1_1_2_1_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> = 
<font color="Maroon">e2</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">e1</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon">e1</font>
<b>by </b><i>, <a class="ref" href="yellow_5.html#T8">YELLOW_5:8</a></i>
;<br/><b>hence </b><a NAME="E3:7_1_1_2_1_1_3"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:7_1_1_2_1_1_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> = 
<font color="Maroon">e2</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">e2</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon">e2</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E3:7_1_1_2_1_1_4"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">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="E2:7_1_1_2"/>
verum
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:7_1_1"/><i><font color="Green">E63</font></i>: 
not  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span> is <a href="realset2.html#V3">trivial</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_3"/>
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span><b> such that </b><br/><a NAME="E3:7_1_1_3"/><i><font color="Green">E65</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_1.html#D1">TEX_1:def 1</a></i>;<br/>
<a NAME="E4:7_1_1_3"/>
 <a href="eqrel_1.html#K1">nabla</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T9">ZFMISC_1:9</a></i>;<br/>
<a NAME="E5:7_1_1_3"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D1">EQREL_1:def 1</a></i>;<br/>
<a NAME="E6:7_1_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:7_1_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:7_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span> is <a href="lattice8.html#V1" target="_self">finitely_typed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_4"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:7_1_1_4"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> holds <br/><font color="Olive">e</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>

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

<b>thus </b><a NAME="E2:7_1_1_4"/>
for <font color="Olive">eq1</font>, <font color="Olive">eq2</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font><br/> for <font color="Olive">x1</font>, <font color="Olive">y1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">eq1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> &amp; <font color="Olive">eq2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">eq1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Olive">eq2</font> holds <br/> ex <font color="Olive">F</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; <font color="Olive">x1</font>,<font color="Olive">y1</font> <a href="lattice5.html#R1">are_joint_by</a> <font color="Olive">F</font>,<font color="Olive">eq1</font>,<font color="Olive">eq2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">eq1</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font>, <font color="Maroon">eq2</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:7_1_1_4_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">eq1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">eq2</font> )
 ;<br/>

<a NAME="E2:7_1_1_4_1"/>
( <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> or <font color="Maroon">eq1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e2</font> )
 
;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:7_1_1_4_1"/><i><font color="Green">E71</font></i>: 
( ( <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> ) or ( <font color="Maroon">eq1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:7_1_1_4_1"/><i><font color="Green">E74</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/>
<a NAME="E7:7_1_1_4_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">z</font>,<font color="Maroon">y1</font></span><a href="finseq_1.html#K11">*&gt;</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="finseq_2.html#T16">FINSEQ_2:16</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1_1_4_1_1"/>
( ( <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> ) or ( <font color="Maroon">eq1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_4_1_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 ;<br/><div class="add"><a NAME="E2:7_1_1_4_1_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E3:7_1_1_4_1_1_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/>( ( not <font color="Olive">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq1</font> ) &amp; ( <font color="Olive">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_4_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_1_4_1_1_1_1"/><i><font color="Green">E78</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:7_1_1_4_1_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E3:7_1_1_4_1_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:7_1_1_4_1_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1_1_4_1_1_1_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> ) )
 </span><b>by </b><i>, , , , , , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_4_1_1_1_1_1_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_1_1_4_1_1_1_1_1_1"/><i><font color="Green">E81</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><a NAME="E4:7_1_1_4_1_1_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:7_1_1_4_1_1_1_1_1_1"/>
( ( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq1</font> ) &amp; ( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq2</font> ) )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#T11">EQREL_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_4_1_1_1_1_1_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_1_1_4_1_1_1_1_1_2"/><i><font color="Green">E83</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><a NAME="E4:7_1_1_4_1_1_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:7_1_1_4_1_1_1_1_1_2"/>
( ( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq1</font> ) &amp; ( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq2</font> ) )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#T11">EQREL_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_4_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:7_1_1_4_1_1_1_1_1_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> )
 ;<br/><div class="add"><b>then </b><font color="Maroon">eq1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">eq2</font> = 
<font color="Maroon">e2</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">e1</font>
<b>by </b><i><a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>
<br/>.= 
<font color="Maroon">eq2</font>
<b>by </b><i>, , <a class="ref" href="yellow_5.html#T8">YELLOW_5:8</a></i>
;<br/><b>hence </b><a NAME="E3:7_1_1_4_1_1_1_1_1_3"/>
( ( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq1</font> ) &amp; ( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq2</font> ) )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_4_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:7_1_1_4_1_1_1_1_1_4"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:7_1_1_4_1_1_1_1_1_4"/>
( ( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq1</font> ) &amp; ( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq2</font> ) )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:7_1_1_4_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="lattice5.html#R1">are_joint_by</a> <font color="Maroon">F</font>,<font color="Maroon">eq1</font>,<font color="Maroon">eq2</font> )
 <b>by </b><i>, <a class="ref" href="lattice5.html#D3">LATTICE5:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_4_1_1_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">eq1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 ;<br/><div class="add"><a NAME="E2:7_1_1_4_1_1_2"/><b>then </b><i><font color="Green">E87</font></i>: 
( <font color="Maroon">eq1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E3:7_1_1_4_1_1_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/>( ( not <font color="Olive">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq1</font> ) &amp; ( <font color="Olive">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_4_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_1_4_1_1_2_1"/><i><font color="Green">E88</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:7_1_1_4_1_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E3:7_1_1_4_1_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:7_1_1_4_1_1_2_1"/><b>then </b><i><font color="Green">E89</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1_1_4_1_1_2_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> ) )
 </span><b>by </b><i>, , , , , , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_4_1_1_2_1_1_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:7_1_1_4_1_1_2_1_1_1"/>
( ( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq1</font> ) &amp; ( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq2</font> ) )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_4_1_1_2_1_1_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 ;<br/><div class="add"><b>then </b><font color="Maroon">eq1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">eq2</font> = 
<font color="Maroon">e1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">e2</font>
<b>by </b><i><a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>
<br/>.= 
<font color="Maroon">eq1</font>
<b>by </b><i>, , <a class="ref" href="yellow_5.html#T8">YELLOW_5:8</a></i>
;<br/><b>hence </b><a NAME="E3:7_1_1_4_1_1_2_1_1_2"/>
( ( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq1</font> ) &amp; ( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq2</font> ) )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_4_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:7_1_1_4_1_1_2_1_1_3"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_1_1_4_1_1_2_1_1_3"/><i><font color="Green">E93</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><a NAME="E4:7_1_1_4_1_1_2_1_1_3"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:7_1_1_4_1_1_2_1_1_3"/>
( ( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq1</font> ) &amp; ( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq2</font> ) )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#T11">EQREL_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_4_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:7_1_1_4_1_1_2_1_1_4"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">eq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">eq2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_1_1_4_1_1_2_1_1_4"/><i><font color="Green">E95</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><a NAME="E4:7_1_1_4_1_1_2_1_1_4"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:7_1_1_4_1_1_2_1_1_4"/>
( ( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq1</font> ) &amp; ( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eq2</font> ) )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#T11">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="E4:7_1_1_4_1_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="lattice5.html#R1">are_joint_by</a> <font color="Maroon">F</font>,<font color="Maroon">eq1</font>,<font color="Maroon">eq2</font> )
 <b>by </b><i>, <a class="ref" href="lattice5.html#D3">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">Y</font> =  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K7">}</a></span> as  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="lattice5.html#NM2">Sublattice</a> of  <a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E13:7_1_1"/>
( not <font color="Maroon">Y</font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Maroon">Y</font> is <a href="lattice8.html#V1" target="_self">finitely_typed</a> &amp; <font color="Maroon">Y</font> is <a href="yellow_0.html#V4">full</a> )
 <b>by </b><i>, </i>;<br/>


</div>
