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









<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">a<sub>1</sub></font> is   <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">A</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">L</font>;<br/>
<a NAME="E1:60"/><i><font color="Green">E29</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="60_1"><b>proof </b></a><div class="add">

<a NAME="E1:60_1"/>
(  <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:60_1"/>
 <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<a href="xboole_0.html#K1">{}</a>  is   <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">A</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>,<font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:60"/><i><font color="Green">E37</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="60_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="E1:60_2"/>
 <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font> is   <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">A</font>,<font color="Maroon">O1</font></span>)</span>,<font color="Maroon">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">CD</font> =  <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font> as   <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">A</font>,<font color="Maroon">O1</font></span>)</span>,<font color="Maroon">L</font> ;<br/>
<a NAME="E3:60_2"/><i><font color="Green">E39</font></i>: 
 <a href="lattice5.html#K9" target="_self">ConsecutiveSet</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> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K6" target="_self">new_set</a> <span class="p1">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span>
 
<b>by </b><i/>;<br/>
 <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</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="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#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</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="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:60"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">CD</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:60_2"><i><font color="Green">E39</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:60_2"/>
 <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</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> is   <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">A</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:60"/><i><font color="Green">E42</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="60_3"><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="E1:60_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">O1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">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="Maroon">O1</font> holds <br/> <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Olive">O2</font> is   <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">A</font>,<font color="Olive">O2</font></span>)</span>,<font color="Maroon">L</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="hidden.html#M1">set</a>  =  <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</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="E3:60_3"/><i><font color="Green">E49</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O1</font> &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="Maroon">O1</font> holds <br/><font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">O2</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E4:60_3"/><i><font color="Green">E50</font></i>: 
 <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">O1</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="lattice5.html#E3:60_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</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="lattice5.html#K9" target="_self">ConsecutiveSet</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="E6:60_3"/><i><font color="Green">E51</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ts</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O1</font> &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="Maroon">O1</font> holds <br/><font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">O2</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<b>set </b><font color="Maroon">CS</font> =  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font>;<br/>
<b>set </b><font color="Maroon">Y</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </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>;<br/>
<a NAME="E7:60_3"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">O</font>, <font color="Olive">O2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">O2</font> &amp; <font color="Olive">O2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> holds <br/><font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_1"><b>proof </b></a><div class="add">

<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>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">O</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> implies <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:60_3_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>2</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="E2:60_3_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">O2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">O2</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">O2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_1_1"><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="E1:60_3_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">O</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">O2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> implies <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O2</font> )
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:60_3_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">O</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>
 <b>and </b><br/><a NAME="E3:60_3_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_3_1_1_1"/>
( <font color="Maroon">O</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font> or <font color="Maroon">O</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:60_3_1_1_1_1"/>
<font color="Maroon">O</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_3_1_1_1_1"/>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:60_3_1_1_1_2"/>
<font color="Maroon">O</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>
 ;<br/><div class="add"><a NAME="E2:60_3_1_1_1_2"/><b>then </b>
<font color="Maroon">O</font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:60_3_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:60_3_1_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:60_3_1_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">O2</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><a NAME="E5:60_3_1_1_1_2"/><b>then </b><i><font color="Green">E60</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">Ls</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:60_3_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon">cd2</font> =  <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">O2</font> as   <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">A</font>,<font color="Maroon">O2</font></span>)</span>,<font color="Maroon">L</font> <b>by </b><i><a class="txt" href="lattice5.html#E3:60_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a></i>;<br/><font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font></span>)</span> = 
 <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="lattice5.html#E3:60_3_1_1_1_2"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
 <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#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">O2</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font></span>)</span>,<font color="Maroon">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">q</font>,<font color="Maroon">O2</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:60"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">cd2</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">q</font>,<font color="Maroon">O2</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E8:60_3_1_1_1_2"/><b>then </b>
 <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">O2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:60_3_1_1_1_2"/><b>then </b>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="lattice5.html#E3:60_3_1"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E10:60_3_1_1_1_2"/>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O2</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:60_3_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="lattice5.html#E3:60_3_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice5.html#E4:60_3_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="lattice5.html#E5:60_3_1_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</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="E3:60_3_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">O2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">O2</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">O3</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O3</font> <a href="hidden.html#R2">in</a> <font color="Olive">O2</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">O3</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">O2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_1_2"><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><b>that </b><br/><a NAME="E1:60_3_1_2"/><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>  )
 <b>and </b><br/><a NAME="E2:60_3_1_2"/>
for <font color="Olive">O3</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">O</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">O3</font> &amp; <font color="Olive">O3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> holds <br/><font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O3</font>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:60_3_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">O</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O2</font>
 <b>and </b><br/><a NAME="E4:60_3_1_2"/><i><font color="Green">E64</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">Ls</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">Lt</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E6:60_3_1_2"/><i><font color="Green">E65</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Lt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O2</font> &amp; ( for <font color="Olive">O3</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> holds <br/><font color="Maroon">Lt</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">O3</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<i><font color="Green">E66</font></i>: <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O2</font> = 
 <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">O2</font>
<b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a>, </i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Lt</font></span>)</span>
<b>by </b><i>, , </i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_3_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_3_1_2_2"/>
( <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O2</font> or <font color="Maroon">O</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">O2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_3_1_2_2_1"><b>suppose </b></a><a NAME="E1:60_3_1_2_2_1"/>
<font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_3_1_2_2_1"/>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_3_1_2_2_2"><b>suppose </b></a><a NAME="E1:60_3_1_2_2_2"/>
<font color="Maroon">O</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">O2</font>
 ;<br/><div class="add"><a NAME="E2:60_3_1_2_2_2"/><b>then </b>
<font color="Maroon">O</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">O2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:60_3_1_2_2_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:60_3_1_2_2_2"/><b>then </b>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><b>then </b><font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O</font> = 
 <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">O</font>
<b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<font color="Maroon">Lt</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O</font>
<b>by </b><i>, <a class="txt" href="lattice5.html#E1:60_3_1_2"><i><font color="Green">E62</font></i></a></i>
;<br/><a NAME="E6:60_3_1_2_2_2"/><b>then </b>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Lt</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E1:60_3_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E7:60_3_1_2_2_2"/>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:60_3_1"><i><font color="Green">E61</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>
<b>thus </b><a NAME="E4:60_3_1"/>
for <font color="Olive">O2</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">O2</font>]
 <b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="lattice5.html#E2:60_3_1"><i><font color="Green">E54</font></i></a>, , );</i><br/>


</div><b>end;</b></div>
<a NAME="E8:60_3"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font> holds <br/><font color="Olive">x</font>,<font color="Olive">y</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:60_3_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">o1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:60_3_2"/><i><font color="Green">E72</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">Ls</font> &amp; <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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">o2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:60_3_2"/><i><font color="Green">E73</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">Ls</font> &amp; <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</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="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E7:60_3_2"/>
( <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> )
 
;<br/>
<a NAME="E8:60_3_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E9:60_3_2"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:60_3"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font> is <a href="ordinal1.html#V6">c=-linear</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E10:60_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:60_3_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font>
 ;<br/>

<b>then consider </b><font color="Maroon">o</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:60_3_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</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">o</font> = <font color="Maroon">o</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:60_3_3"/>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> as   <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">A</font>,<font color="Maroon">o</font></span>)</span>,<font color="Maroon">L</font> <b>by </b><i><a class="txt" href="lattice5.html#E3:60_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<a NAME="E7:60_3_3"/><i><font color="Green">E76</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</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="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o</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="E8:60_3_3"/><i><font color="Green">E77</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<a NAME="E9:60_3_3"/>
<font color="Maroon">o</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E10:60_3_3"/><b>then </b>
 <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:60_3_3"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</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="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E12:60_3_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:60_3"/><b>then </b>
 <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> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="hahnban.html#T13">HAHNBAN:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E13:60_3"/><i><font color="Green">E78</font></i>: 
(  <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> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<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="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/>
<a NAME="E15:60_3"/>
<font color="Maroon">Ls</font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:60_3_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:60_3_4"/>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:60_3_4"/>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:60_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">LsF</font> = <font color="Maroon">Ls</font> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E17:60_3"/><i><font color="Green">E81</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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> <span class="p2">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">LsF</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">O1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="lattice5.html#E3:60_3"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">YY</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">O2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font> : verum </span>}</span> ;<br/>
<a NAME="E19:60_3"/><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">Ls</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">O2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:60_3_5"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">Ls</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">O2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font> : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:60_3_5_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">Ls</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">o</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:60_3_5_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">Ls</font></span>)</span> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">Ls</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:60_3_5_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">LsF</font>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o'</font> = <font color="Maroon">o</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font> <b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E6:60_3_5_1"/>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o'</font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">o'</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ls</font> = <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o'</font> as   <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">A</font>,<font color="Maroon">o'</font></span>)</span>,<font color="Maroon">L</font> <b>by </b><i><a class="txt" href="lattice5.html#E3:60_3"><i><font color="Green">E49</font></i></a></i>;<br/>
<font color="Maroon">Z</font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">ls</font>
<b>by </b><i>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o'</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o'</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/>
<b>hence </b><a NAME="E9:60_3_5_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">O2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font> : verum </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:60_3_5"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">O2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">o</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font><b> such that </b><br/><a NAME="E4:60_3_5"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">Z</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="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>
<a NAME="E5:60_3_5"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">LsF</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E6:60_3_5"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">LsF</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<a NAME="E7:60_3_5"/>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ls</font> = <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> as   <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">A</font>,<font color="Maroon">o</font></span>)</span>,<font color="Maroon">L</font> <b>by </b><i><a class="txt" href="lattice5.html#E3:60_3"><i><font color="Green">E49</font></i></a></i>;<br/>
<font color="Maroon">Z</font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">ls</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">Ls</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="txt" href="lattice5.html#E4:60_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
;<br/>
<b>hence </b><a NAME="E10:60_3_5"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">Ls</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:60_3"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">O1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:60_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">RTs</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ts</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="lattice5.html#E7:60_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E22:60_3"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RTs</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RTs</font> holds <br/><font color="Olive">x</font>,<font color="Olive">y</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:60_3_6"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RTs</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RTs</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">o1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:60_3_6"/><i><font color="Green">E92</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">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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">o2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:60_3_6"/><i><font color="Green">E93</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">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</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="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E7:60_3_6"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1'</font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o1'</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:60_3"><i><font color="Green">E52</font></i></a>, </i>;<br/>
<a NAME="E8:60_3_6"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2'</font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o2'</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:60_3"><i><font color="Green">E52</font></i></a>, </i>;<br/>
<a NAME="E9:60_3_6"/>
( <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> )
 
;<br/>
<a NAME="E10:60_3_6"/><b>then </b>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E11:60_3_6"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:60_3"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">RTs</font> is <a href="ordinal1.html#V6">c=-linear</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E24:60_3"/><i><font color="Green">E98</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">O2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">RTs</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RTs</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_7"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">XX</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">RTs</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RTs</font> </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:60_3_7"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">O2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">RTs</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RTs</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:60_3_7_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">O2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">o</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font><b> such that </b><br/><a NAME="E3:60_3_7_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">Z</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="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>
<a NAME="E4:60_3_7_1"/>
<font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:60_3"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">CoS</font> =  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">RTs</font> <b>by </b><i><a class="txt" href="lattice5.html#E7:60_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:60_3_7_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">CoS</font>,<font color="Maroon">CoS</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:60_3_7_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">RTs</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RTs</font> </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:60_3_7"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">RTs</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RTs</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">RTs</font><b> such that </b><br/><a NAME="E4:60_3_7"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">Z</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> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RTs</font> )
 ;<br/>
<b>consider </b><font color="Maroon">o</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:60_3_7"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">o</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">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</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">o'</font> = <font color="Maroon">o</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E8:60_3_7"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">o'</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:60_3"><i><font color="Green">E52</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">O</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font><b> such that </b><br/><a NAME="E10:60_3_7"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E7:60_3"><i><font color="Green">E52</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E11:60_3_7"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Olive">O2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">O2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o1</font> : verum </span>}</span> 
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Ts</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</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="txt" href="lattice5.html#E3:60_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice5.html#E7:60_3"><i><font color="Green">E52</font></i></a>, </i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">RTs</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">RTs</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:60_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice5.html#E7:60_3"><i><font color="Green">E52</font></i></a>, </i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
<b>by </b><i>, , , , </i>
;<br/>
<b>hence </b><a NAME="E26:60_3"/>
 <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font> is   <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">A</font>,<font color="Maroon">O1</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E6:60_3"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:60"/>
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="E5:60"/>
 <a href="lattice5.html#K12" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<font color="Maroon">O</font> is   <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">A</font>,<font color="Maroon">O</font></span>)</span>,<font color="Maroon">L</font>
 ;<br/>


</div>
