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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">d</font> is <a href="lattice5.html#V1">symmetric</a>
 ;<br/>

<b>let </b><font color="Maroon">q</font> be    <a href="lattice5.html#M1">QuadrSeq</a> of <font color="Maroon">d</font>;<br/><b>let </b><font color="Maroon">O</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">a<sub>1</sub></font> is <a href="lattice5.html#V1">symmetric</a>;<br/>
<a NAME="E2:41"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<a href="xboole_0.html#K1">{}</a> , <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<a href="xboole_0.html#K1">{}</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y'</font>,<font color="Maroon">x'</font>
<b>by </b><i>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:41"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">O1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">O1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:41_2"/>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font> is <a href="lattice5.html#V1">symmetric</a>
 ;<br/>

<a NAME="E2:41_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span> is <a href="lattice5.html#V1">symmetric</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E60</font></i>:  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span> = 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11">BiFun</a> <span class="p2">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span>,<font color="Maroon">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D16">LATTICE5:def 16</a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">y'</font>,<font color="Maroon">x'</font>
<b>by </b><i>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E4:41"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">O1</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">O2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O2</font> <a href="hidden.html#R2">in</a> <font color="Olive">O1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O2</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">O2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:41_3"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">O2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">O2</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> holds <br/> <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Olive">O1</font> is <a href="lattice5.html#V1">symmetric</a> ) )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>  <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">L</font> =  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">Ls</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:41_3"/><i><font color="Green">E63</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O2</font> &amp; ( for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> holds <br/><font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">O1</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E4:41_3"/><i><font color="Green">E65</font></i>: 
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O2</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="lattice8.html#E1:41"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">Ts</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E6:41_3"/><i><font color="Green">E66</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ts</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O2</font> &amp; ( for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> holds <br/><font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">O1</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E7:41_3"/><i><font color="Green">E69</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Ts</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">CS</font> =  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font>;<br/>
<b>reconsider </b><font color="Maroon">f</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font></span>)</span> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font></span>)</span>,<font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E9:41_3"/>
<font color="Maroon">f</font> is <a href="lattice5.html#V1">symmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a><br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:41_3_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ts</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E2:41"><i><font color="Green">E42</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">o1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:41_3_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ts</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:41_3_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ts</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E2:41"><i><font color="Green">E42</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">o2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:41_3_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ts</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">o1</font>, <font color="Maroon">o2</font> = <font color="Maroon">o2</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="lattice8.html#D2" target="_self">Def2</a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E10:41_3_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:41_3_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>
 
<b>by </b><i>, , <a class="ref" href="lattice8.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E12:41_3_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:41"><i><font color="Green">E39</font></i></a>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font></span>)</span>,<font color="Maroon">L</font> ;<br/>
<a NAME="E14:41_3_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">h1</font> is <a href="lattice5.html#V1">symmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a><br/>

<a NAME="E1:41_3_1_1"/><i><font color="Green">E82</font></i>: 
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o1</font> is <a href="lattice5.html#V1">symmetric</a>
 
<b>by </b><i>, , </i>;<br/>
<b>thus </b><font color="Maroon">h1</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o1</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:41"><i><font color="Green">E39</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o1</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>
<br/>.= 
<font color="Maroon">h1</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:41"><i><font color="Green">E39</font></i></a>, , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E15:41_3_1"/><i><font color="Green">E83</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:41_3_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o2</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:41"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="lattice8.html#D2" target="_self">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h2</font> = <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font> ;<br/>
<a NAME="E18:41_3_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">h2</font> is <a href="lattice5.html#V1">symmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a><br/>

<a NAME="E1:41_3_1_2"/><i><font color="Green">E86</font></i>: 
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o2</font> is <a href="lattice5.html#V1">symmetric</a>
 
<b>by </b><i>, , <a class="ref" href="lattice8.html#D2" target="_self">Def2</a></i>;<br/>
<b>thus </b><font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o2</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:41"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="lattice8.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o2</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>
<br/>.= 
<font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:41"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="lattice8.html#D2" target="_self">Def2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E19:41_3_1"/><i><font color="Green">E87</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_3_1_3"/>
( <font color="Maroon">o1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o2</font> or <font color="Maroon">o2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1_3_1"><b>suppose </b></a><a NAME="E1:41_3_1_3_1"/>
<font color="Maroon">o1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o2</font>
 ;<br/><div class="add"><a NAME="E2:41_3_1_3_1"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font> <b>by </b><i>, , <a class="ref" href="lattice8.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E4:41_3_1_3_1"/>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:41"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="lattice8.html#D2" target="_self">Def2</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:41_3_1_3_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">h2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E6:41_3_1_3_1"/><i><font color="Green">E90</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font> )
 <b>by </b><i>, , , <a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>
<b>by </b><i><a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y'</font>,<font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1_3_2"><b>suppose </b></a><a NAME="E1:41_3_1_3_2"/>
<font color="Maroon">o2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><a NAME="E2:41_3_1_3_2"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font> <b>by </b><i>, , , </i>;<br/><a NAME="E4:41_3_1_3_2"/>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:41"><i><font color="Green">E39</font></i></a>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:41_3_1_3_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">h1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E6:41_3_1_3_2"/><i><font color="Green">E93</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font> )
 <b>by </b><i>, , , <a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">h1</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>
<b>by </b><i><a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">h1</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y'</font>,<font color="Maroon">x'</font>
<b>by </b><i>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E10:41_3"/>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O2</font> is <a href="lattice5.html#V1">symmetric</a>
 <b>by </b><i>, <a class="txt" href="lattice8.html#E1:41"><i><font color="Green">E39</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:41"/>
for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(, , );<br/></i>
<b>hence </b><a NAME="E6:41"/>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O</font> is <a href="lattice5.html#V1">symmetric</a>
 ;<br/>


</div>
