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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="lattice5.html#M1">QuadrSeq</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>









<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means  <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:36"/><i><font color="Green">E30</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="36_1"><b>proof </b></a><div class="add">

<a NAME="E1:36_1"/>
(  <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="lattice8.html#T15" target="_self">Th15</a>, <a class="ref" href="lattice8.html#T19" target="_self">Th19</a></i>;<br/>
<b>hence </b><a NAME="E2:36_1"/>
 <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:36"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:36_2"/>
 <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> is   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E3:36_2"/><i><font color="Green">E32</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K1" target="_self">new_set2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice8.html#T16" target="_self">Th16</a></i>;<br/>
 <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11">BiFun</a> <span class="p2">(<span class="default"><a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#T20" target="_self">Th20</a></i>
<br/>.= 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D16">LATTICE5:def 16</a></i>
;<br/>
<b>hence </b><a NAME="E5:36_2"/>
 <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> is   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:36_2"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:36"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:36_3"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/> <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> is   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:36_3"/><i><font color="Green">E34</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E4:36_3"/><i><font color="Green">E35</font></i>: 
 <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:36_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="lattice8.html#T21" target="_self">Th21</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E6:36_3"/><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></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">c<sub>7</sub></font></span>)</span>;<br/>
<a NAME="E7:36_3"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></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>] means ( <font color="Maroon">c<sub>13</sub></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">c<sub>7</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:36_3_1"/><i><font color="Green">E38</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="E2:36_3_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:36_3_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:36_3_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E3:36_3_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_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="E1:36_3_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_3_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_3_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_3_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_3_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:36_3_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:36_3_1_1_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:36_3_1_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><a NAME="E5:36_3_1_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:36_3_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="lattice8.html#E1:36_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a></i>;<br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
 <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3_1_1"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11">BiFun</a> <span class="p2">(<span class="default"><a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#T20" target="_self">Th20</a></i>
<br/>.= 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D16">LATTICE5:def 16</a></i>
;<br/><a NAME="E8:36_3_1_1_1_2"/><b>then </b>
 <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice8.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E9:36_3_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lattice8.html#E5:36_3_1_1_1_2"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E10:36_3_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:36_3_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="lattice8.html#E4:36_3_1_1_1_2"><i><font color="Green">E44</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="E3:36_3_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:36_3_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:36_3_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:36_3_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E4:36_3_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></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="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E6:36_3_1_2"/><i><font color="Green">E44</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<i><font color="Green">E45</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
 <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lattice8.html#E4:36_3_1_2"><i><font color="Green">E43</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">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:36_3_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="lattice8.html#E6:36_3_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattice8.html#T21" target="_self">Th21</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_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="E1:36_3_1_2_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1_2_2_1"><b>suppose </b></a><a NAME="E1:36_3_1_2_2_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_3_1_2_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_3_1_2_2_2"><b>suppose </b></a><a NAME="E1:36_3_1_2_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_3_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:36_3_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:36_3_1_2_2_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:36_3_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lattice8.html#E4:36_3_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><b>then </b><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
 <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E6:36_3_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3_1_2_2_2"><i><font color="Green">E46</font></i></a></i>
;<br/><a NAME="E6:36_3_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E6:36_3_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3_1_2_2_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E7:36_3_1_2_2_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:36_3_1_2"><i><font color="Green">E45</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="E4:36_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="lattice8.html#E1:36_3_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="lattice8.html#E2:36_3_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3_1"><i><font color="Green">E40</font></i></a>);</i><br/>


</div><b>end;</b></div>
<a NAME="E8:36_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_3_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_3_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></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">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:36_3_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:36_3_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="lattice8.html#E3:36_3_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="lattice8.html#E5:36_3_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E7:36_3_2"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>18</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> )
 
;<br/>
<a NAME="E8:36_3_2"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:36_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="lattice8.html#E5:36_3_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:36_3_2"/>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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="E9:36_3"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></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="E10:36_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></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="E1:36_3_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_3_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></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">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="lattice8.html#E3:36_3_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:36_3_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3_3"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="lattice8.html#E1:36_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3_3"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E7:36_3_3"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></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="E8:36_3_3"/><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E9:36_3_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E10:36_3_3"/><b>then </b>
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="lattice8.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E11:36_3_3"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:36_3_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E12:36_3_3"/>
<font color="Maroon">c<sub>13</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:36_3_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="lattice8.html#E8:36_3_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:36_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">c<sub>7</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E9:36_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hahnban.html#T13">HAHNBAN:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E13:36_3"/><i><font color="Green">E39</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">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></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">c<sub>7</sub></font></span>)</span> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="lattice8.html#E13:36_3"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E15:36_3"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></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="E1:36_3_4"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></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="E3:36_3_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lattice8.html#E1:36_3_4"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:36_3_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:36_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lattice8.html#E1:36_3_4"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E17:36_3"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice5.html#T1">LATTICE5:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>6</sub></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="lattice8.html#E1:36_3"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font> : verum </span>}</span> ;<br/>
<a NAME="E19:36_3"/><i><font color="Green">E41</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">c<sub>7</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_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">c<sub>7</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></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="36_3_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></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="E1:36_3_5_1"/>
<font color="Maroon">c<sub>18</sub></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">c<sub>7</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_3_5_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>19</sub></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">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:36_3_5_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3_5_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font> <b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E6:36_3_5_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="lattice8.html#E1:36_3"><i><font color="Green">E33</font></i></a></i>;<br/>
<font color="Maroon">c<sub>18</sub></font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3_5_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="lattice8.html#E4:36_3_5_1"><i><font color="Green">E43</font></i></a>, <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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>20</sub></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="E9:36_3_5_1"/>
<font color="Maroon">c<sub>18</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font> : verum </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>18</sub></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="E2:36_3_5"/>
<font color="Maroon">c<sub>18</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E4:36_3_5"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>
<a NAME="E5:36_3_5"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E6:36_3_5"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>19</sub></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">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<a NAME="E7:36_3_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="lattice8.html#E1:36_3"><i><font color="Green">E33</font></i></a></i>;<br/>
<font color="Maroon">c<sub>18</sub></font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E4:36_3_5"><i><font color="Green">E42</font></i></a>, <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">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
;<br/>
<b>hence </b><a NAME="E10:36_3_5"/>
<font color="Maroon">c<sub>18</sub></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">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E6:36_3_5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:36_3"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:36_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></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="lattice8.html#E6:36_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E22:36_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_3_6"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_3_6"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></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">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:36_3_6"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:36_3_6"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="lattice8.html#E3:36_3_6"><i><font color="Green">E43</font></i></a>, <a class="txt" href="lattice8.html#E5:36_3_6"><i><font color="Green">E44</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E7:36_3_6"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E6:36_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="lattice8.html#E3:36_3_6"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E8:36_3_6"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E6:36_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="lattice8.html#E5:36_3_6"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E9:36_3_6"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>24</sub></font> or <font color="Maroon">c<sub>24</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>23</sub></font> )
 
;<br/>
<a NAME="E10:36_3_6"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3_6"><i><font color="Green">E43</font></i></a>, <a class="txt" href="lattice8.html#E5:36_3_6"><i><font color="Green">E44</font></i></a>, <a class="txt" href="lattice8.html#E7:36_3_6"><i><font color="Green">E45</font></i></a>, <a class="txt" href="lattice8.html#E8:36_3_6"><i><font color="Green">E46</font></i></a>, <a class="ref" href="lattice8.html#T22" target="_self">Th22</a></i>;<br/>
<b>hence </b><a NAME="E11:36_3_6"/>
<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></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="E23:36_3"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>18</sub></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="E24:36_3"/><i><font color="Green">E43</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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>18</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_7"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>18</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:36_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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>18</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></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="36_3_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></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="E1:36_3_7_1"/>
<font color="Maroon">c<sub>20</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E3:36_3_7_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>
<a NAME="E4:36_3_7_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E6:36_3"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> =  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>21</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>18</sub></font> <b>by </b><i><a class="txt" href="lattice8.html#E6:36_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:36_3_7_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:36_3_7_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:36_3_7_1"/>
<font color="Maroon">c<sub>20</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>18</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>20</sub></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="E2:36_3_7"/>
<font color="Maroon">c<sub>20</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>18</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>18</sub></font><b> such that </b><br/><a NAME="E4:36_3_7"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:36_3_7"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></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">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="lattice8.html#E6:36_3_7"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E8:36_3_7"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E6:36_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="lattice8.html#E6:36_3_7"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E10:36_3_7"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E6:36_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="lattice8.html#E6:36_3_7"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:36_3_7"/>
<font color="Maroon">c<sub>20</sub></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="lattice8.html#E4:36_3_7"><i><font color="Green">E44</font></i></a>, <a class="txt" href="lattice8.html#E8:36_3_7"><i><font color="Green">E46</font></i></a>, <a class="txt" href="lattice8.html#E10:36_3_7"><i><font color="Green">E47</font></i></a></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="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:36_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="lattice8.html#E6:36_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="lattice8.html#T17" target="_self">Th17</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">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:36_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="lattice8.html#E6:36_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="lattice8.html#T17" target="_self">Th17</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E17:36_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="lattice8.html#E19:36_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="lattice8.html#E23:36_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="lattice8.html#E24:36_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="lattice5.html#T3">LATTICE5:3</a></i>
;<br/>
<b>hence </b><a NAME="E26:36_3"/>
 <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> is   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:36_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="lattice8.html#E13:36_3"><i><font color="Green">E39</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="E4:36"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="lattice8.html#E1:36"><i><font color="Green">E30</font></i></a>, <a class="txt" href="lattice8.html#E2:36"><i><font color="Green">E31</font></i></a>, <a class="txt" href="lattice8.html#E3:36"><i><font color="Green">E32</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:36"/>
 <a href="lattice8.html#K5" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> is   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
