<?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> ;<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>;<br/><b>let </b><font color="Maroon" title="c3">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>;<br/>





<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c3">d</font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">q</font> be    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c3">d</font>;<br/><b>let </b><font color="Maroon" title="c5">O</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<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> <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,$1 is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>;<br/>
<a NAME="E2:39"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">z'</font> = <font color="Maroon" title="c6">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="lattice8.html#T15" target="_self" title="LATTICE8:th.15">Th15</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c6">z</font> = 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">z'</font>,<font color="Maroon" title="c7">z'</font>
<b>by </b><i><a class="ref" href="lattice8.html#T19" target="_self" title="LATTICE8:th.19">Th19</a></i>
<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:39"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:39"/><i><font color="Green" title="E34">A3</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="39_2"><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>;<br/>

<b>assume </b><a NAME="E1:39_2"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>
 ;<br/>

<a NAME="E2:39_2"/><b>then </b><i><font color="Green" title="E35">A4</font></i>: 
 <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>
 
<b>by </b><i><a class="ref" href="lattice8.html#T10" target="_self" title="LATTICE8:th.10">Th10</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a><br/>

<b>reconsider </b><font color="Maroon" title="c8">z'</font> = <font color="Maroon" title="c7">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <b>by </b><i><a class="ref" href="lattice8.html#T16" target="_self" title="LATTICE8:th.16">Th16</a></i>;<br/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</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="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</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="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#T20" target="_self" title="LATTICE8:th.20">Th20</a></i>
<br/>.= 
 <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D16" title="LATTICE5:def.16">LATTICE5:def 16</a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c7">z</font> = 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c8">z'</font>,<font color="Maroon" title="c8">z'</font>

<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="txt" href="lattice8.html#E2:39_2"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E4:39"/><i><font color="Green" title="E35">A5</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="Olive" title="b1">O1</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">O1</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">O2</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><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>;<br/>

<b>assume </b><a NAME="E1:39_3"/><i><font color="Green" title="E36">A6</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/> <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">O1</font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a> ) )
 ;<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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,$1</span>)</span>,<font color="Maroon" title="c2">L</font> =  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">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="E3:39_3"/><i><font color="Green" title="E37">A7</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="E4:39_3"/><i><font color="Green" title="E38">A8</font></i>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">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="lattice8.html#E1:39_3"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="lattice8.html#E3:39_3"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th21</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,$1;<br/>
<b>consider </b><font color="Maroon" title="c8">Ts</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E6:39_3"/><i><font color="Green" title="E39">A9</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">Ts</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="c8">Ts</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>2</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="E7:39_3"/><i><font color="Green" title="E40">A10</font></i>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</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="c8">Ts</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:39_3"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="lattice8.html#E6:39_3"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="lattice8.html#T17" target="_self" title="LATTICE8:th.17">Th17</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">CS</font> =  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f</font> =  <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> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font></span>)</span>,<font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="lattice8.html#E4:39_3"><i><font color="Green" title="E38">A8</font></i></a></i>;<br/>
<a NAME="E9:39_3"/>
<font color="Maroon" title="c10">f</font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a><br/>

<b>consider </b><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:39_3_1"/><i><font color="Green" title="E41">A11</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c12">y</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="c8">Ts</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E7:39_3"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:39_3_1"/><i><font color="Green" title="E42">A12</font></i>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">Ts</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E2:39_3_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">o</font> = <font color="Maroon" title="c13">o</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="lattice8.html#E4:39_3_1"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E6:39_3_1"/><i><font color="Green" title="E43">A13</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c14">o</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E6:39_3"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="lattice8.html#E2:39_3_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="lattice8.html#E4:39_3_1"><i><font color="Green" title="E42">A12</font></i></a></i>;<br/>
<a NAME="E7:39_3_1"/><i><font color="Green" title="E44">A14</font></i>: 
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">o</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:39_3"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="lattice8.html#E6:39_3"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="lattice8.html#E4:39_3_1"><i><font color="Green" title="E42">A12</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">h</font> = <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">o</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c14">o</font></span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E9:39_3_1"/><i><font color="Green" title="E45">A15</font></i>: 
<font color="Maroon" title="c15">h</font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c14">o</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a><br/>

<a NAME="E1:39_3_1_1"/><i><font color="Green" title="E46">A16</font></i>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">o</font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:39_3"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="lattice8.html#E6:39_3"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="lattice8.html#E4:39_3_1"><i><font color="Green" title="E42">A12</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c15">h</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c16">z</font> = 
<span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">o</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c16">z</font>
<b>by </b><i><a class="txt" href="lattice8.html#E3:39_3"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="lattice8.html#E6:39_3"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="lattice8.html#E4:39_3_1"><i><font color="Green" title="E42">A12</font></i></a></i>
<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:39_3_1_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:39_3_1"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">o</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="lattice8.html#E3:39_3"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="lattice8.html#E6:39_3"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="lattice8.html#E4:39_3_1"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="lattice8.html#E7:39_3_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:39_3_1"/><b>then </b><i><font color="Green" title="E46">A17</font></i>: 
<font color="Maroon" title="c15">h</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c10">f</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:39_3_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">x'</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c14">o</font> <b>by </b><i><a class="txt" href="lattice8.html#E6:39_3"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="lattice8.html#E2:39_3_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="lattice8.html#E4:39_3_1"><i><font color="Green" title="E42">A12</font></i></a></i>;<br/>
<a NAME="E13:39_3_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c14">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c14">o</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:39_3_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">h</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E6:39_3_1"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</font> = 
<font color="Maroon" title="c15">h</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">x'</font>,<font color="Maroon" title="c16">x'</font>
<b>by </b><i><a class="txt" href="lattice8.html#E11:39_3_1"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="txt" href="lattice8.html#E9:39_3_1"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:39_3"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O2</font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:39_3"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="lattice8.html#E3:39_3"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:39"/>
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="lattice8.html#E2:39"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="lattice8.html#E3:39"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="lattice8.html#E4:39"><i><font color="Green" title="E35">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:39"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">O</font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>
 ;<br/>


</div>
