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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="lattice5.html#M1" target="_self">QuadrSeq</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>











<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ( <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> implies  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:62"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E2:62"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> implies  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:62_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1"><b>suppose </b></a><a NAME="E1:62_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_1_1_1"/>
 <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2"><b>suppose </b></a><a NAME="E1:62_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:62_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:62_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:62_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/> <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11" target="_self">BiFun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#T30" target="_self">Th30</a></i>
<br/>.= 
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D16" target="_self">Def16</a></i>
;<br/><a NAME="E5:62_1_1_2"/><b>then </b>
 <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice5.html#T22" target="_self">Th22</a></i>;<br/><b>hence </b><a NAME="E6:62_1_1_2"/>
 <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:62_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="lattice5.html#E3:62_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:62"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:62_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:62_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<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="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> =  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E4:62_2"/><i><font color="Green">E53</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E5:62_2"/><i><font color="Green">E54</font></i>: 
 <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:62_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="lattice5.html#E4:62_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="lattice5.html#T31" target="_self">Th31</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1_1"><b>suppose </b></a><a NAME="E1:62_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_2_1_1"/>
 <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1_2"><b>suppose </b></a><a NAME="E1:62_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:62_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:62_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:62_2_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:62_2_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:62_2"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E5:62_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:62_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="lattice5.html#E3:62_2_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:62_2_1_2"/>
 <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E5:62_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="lattice5.html#E4:62_2_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:62"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="lattice5.html#E1:62"><i><font color="Green">E48</font></i></a>, <a class="txt" href="lattice5.html#E2:62"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice5.html#E3:62"><i><font color="Green">E50</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:62"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> implies  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>


</div>
