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

<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a><br/> for <font color="Olive" title="b2">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b3">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">L</font><br/> for <font color="Olive" title="b4">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b3">d</font> holds  <font color="Olive" title="b3">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Olive" title="b4">q</font>,<font color="Olive" title="b2">O</font></span><br/><b>let </b><font color="Maroon" title="c2">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b3">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b2">d</font> holds  <font color="Olive" title="b2">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Olive" title="b3">q</font>,<font color="Olive" title="b1">O</font></span><br/><b>let </b><font color="Maroon" title="c3">O</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b1">d</font> holds  <font color="Olive" title="b1">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c3">O</font></span><br/><b>let </b><font color="Maroon" title="c4">d</font> be   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c4">d</font> holds  <font color="Maroon" title="c4">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c3">O</font></span><br/><b>let </b><font color="Maroon" title="c5">q</font> be    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c4">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">O</font></span><br/>









<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b><font color="Maroon" title="c4">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,$1;<br/>
<a NAME="E1:62"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="lattice5.html#T29" target="_self" title="LATTICE5:th.29">Th29</a></i>;<br/>
<a NAME="E2:62"/><i><font color="Green" title="E47">A2</font></i>: 
for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">O1</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" title="c6">O1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">O1</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font>] )</span><br/>

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font>]</span><br/>

 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span> = 
 <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11" title="LATTICE5:func.11">BiFun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<font color="Maroon" title="c2">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" title="LATTICE5:func.10">Quadr</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#T30" target="_self" title="LATTICE5:th.30">Th30</a></i>
<br/>.= 
 <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" title="LATTICE5:func.10">Quadr</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D16" target="_self" title="LATTICE5:def.16">Def16</a></i>
;<br/>
<a NAME="E3:62_1"/><b>then </b>
 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice5.html#T22" target="_self" title="LATTICE5:th.22">Th22</a></i>;<br/>
<b>hence </b><a NAME="E4:62_1"/>
<font color="Maroon" title="c4">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:62_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:62"/><i><font color="Green" title="E48">A4</font></i>: 
for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">O2</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">O1</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O2</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" title="c6">O2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O1</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">O2</font>] )</span><br/>

<b>assume </b><a NAME="E1:62_2"/><i><font color="Green" title="E49">A5</font></i>: 
( <font color="Maroon" title="c6">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font> holds <br/><font color="Maroon" title="c4">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Olive" title="b1">O1</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">O2</font>]</span><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,$1</span>)</span>,<font color="Maroon" title="c2">L</font> =  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,$1;<br/>
<b>consider </b><font color="Maroon" title="c7">Ls</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E3:62_2"/><i><font color="Green" title="E50">A6</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</font> &amp; ( for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font> holds <br/><font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">O1</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E4:62_2"/><i><font color="Green" title="E51">A7</font></i>: 
 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Ls</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:62_2"><i><font color="Green" title="E49">A5</font></i></a>, <a class="txt" href="lattice5.html#E3:62_2"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="lattice5.html#T31" target="_self" title="LATTICE5:th.31">Th31</a></i>;<br/>
<a NAME="E5:62_2"/><i><font color="Green" title="E52">A8</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:62_2"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/>
<b>then </b><font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  = 
 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="lattice5.html#E3:62_2"><i><font color="Green" title="E50">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">d</font>
<b>by </b><i><a class="ref" href="lattice5.html#T29" target="_self" title="LATTICE5:th.29">Th29</a></i>
;<br/>
<a NAME="E7:62_2"/><b>then </b>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Ls</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:62_2"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="lattice5.html#E5:62_2"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:62_2"/>
<font color="Maroon" title="c4">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:62_2"><i><font color="Green" title="E51">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:62"/>
for <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="lattice5.html#E1:62"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="lattice5.html#E2:62"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="lattice5.html#E3:62"><i><font color="Green" title="E48">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:62"/>
<font color="Maroon" title="c4">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
