<?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#NM4">distance_function</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  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>5</sub></font> be   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:48"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="lattice5.html#M1">QuadrSeq</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:48"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K7" target="_self">NextSet2</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:48"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K8" target="_self">NextDelta2</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="lattice8.html#D11" target="_self">Def11</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E5:48"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E6:48"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K5">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>4</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="lattice5.html#D14">LATTICE5:def 14</a></i>;<br/>
<a NAME="E7:48"/><b>then </b>
<span class="p1"><a href="domain_1.html#K5">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K5">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E5:48"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:48"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K5">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K5">]</a></span> )
 <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>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="lattice8.html#E9:48"><i><font color="Green">E40</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E11:48"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E9:48"><i><font color="Green">E40</font></i></a>, <a class="ref" href="lattice8.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E12:48"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K8">`1</a> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E9:48"><i><font color="Green">E40</font></i></a>, <a class="ref" href="mcart_1.html#T78">MCART_1:78</a></i>;<br/>
<a NAME="E13:48"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K9">`2</a> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E9:48"><i><font color="Green">E40</font></i></a>, <a class="txt" href="lattice8.html#E11:48"><i><font color="Green">E41</font></i></a>, <a class="ref" href="mcart_1.html#T78">MCART_1:78</a></i>;<br/>
<a NAME="E14:48"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K10">`3</a> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E9:48"><i><font color="Green">E40</font></i></a>, <a class="txt" href="lattice8.html#E11:48"><i><font color="Green">E41</font></i></a>, <a class="ref" href="mcart_1.html#T78">MCART_1:78</a></i>;<br/>
<a NAME="E15:48"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K11">`4</a> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E9:48"><i><font color="Green">E40</font></i></a>, <a class="txt" href="lattice8.html#E11:48"><i><font color="Green">E41</font></i></a>, <a class="ref" href="mcart_1.html#T78">MCART_1:78</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>13</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K4">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font> as   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E19:48"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="ref" href="lattice8.html#T18" target="_self">Th18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>13</sub></font> ;<br/>
<a NAME="E21:48"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="ref" href="lattice8.html#T13" target="_self">Th13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">c<sub>13</sub></font> ;<br/>
<a NAME="E23:48"/><i><font color="Green">E46</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>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="lattice8.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E24:48"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E9:48"><i><font color="Green">E40</font></i></a>, <a class="ref" href="lattice5.html#T28">LATTICE5:28</a></i>;<br/>
<a NAME="E25:48"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>12</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E26:48"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice8.html#T25" target="_self">Th25</a></i>;<br/>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>12</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#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</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>12</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>6</sub></font>,<font color="Maroon">c<sub>12</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>,<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="lattice5.html#D16">LATTICE5:def 16</a></i>
;<br/>
<a NAME="E28:48"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:48"><i><font color="Green">E38</font></i></a>, <a class="txt" href="lattice8.html#E26:48"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E29:48"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E30:48"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E31:48"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">c<sub>13</sub></font>
 
;<br/>
<a NAME="E32:48"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E33:48"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E34:48"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">c<sub>13</sub></font>
 
;<br/>
<a NAME="E35:48"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is <a href="lattice5.html#V2">zeroed</a>
 
<b>by </b><i><a class="ref" href="lattice8.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E36:48"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <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#K1" target="_self">new_set2</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">c<sub>13</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="E37:48"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E31:48"><i><font color="Green">E50</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E38:48"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E31:48"><i><font color="Green">E50</font></i></a>, <a class="txt" href="lattice8.html#E34:48"><i><font color="Green">E51</font></i></a>, <a class="txt" href="lattice8.html#E36:48"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E39:48"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E34:48"><i><font color="Green">E51</font></i></a>, <a class="txt" href="lattice8.html#E36:48"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E40:48"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="lattice8.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E41:48"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</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>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></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="E42:48"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<i><font color="Green">E59</font></i>: <font color="Maroon">c<sub>15</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E40:48"><i><font color="Green">E57</font></i></a>, <a class="txt" href="lattice8.html#E42:48"><i><font color="Green">E58</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>

;<br/>
<a NAME="E44:48"/>
 <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">c<sub>13</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>,<span class="p1">(<span class="default"><a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E23:48"><i><font color="Green">E46</font></i></a>, <a class="txt" href="lattice8.html#E25:48"><i><font color="Green">E47</font></i></a>, <a class="ref" href="lattice8.html#T22" target="_self">Th22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="lattice8.html#E3:48"><i><font color="Green">E37</font></i></a>, <a class="txt" href="lattice8.html#E31:48"><i><font color="Green">E50</font></i></a>, <a class="txt" href="lattice8.html#E34:48"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>20</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>21</sub></font>
;<br/>

<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>20</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E28:48"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice8.html#E37:48"><i><font color="Green">E54</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E12:48"><i><font color="Green">E42</font></i></a>, <a class="txt" href="lattice8.html#E14:48"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E35:48"><i><font color="Green">E52</font></i></a>, <a class="ref" href="lattice5.html#D7">LATTICE5:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E28:48"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice8.html#E38:48"><i><font color="Green">E55</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E12:48"><i><font color="Green">E42</font></i></a>, <a class="txt" href="lattice8.html#E13:48"><i><font color="Green">E43</font></i></a>, <a class="txt" href="lattice8.html#E14:48"><i><font color="Green">E44</font></i></a>, <a class="txt" href="lattice8.html#E15:48"><i><font color="Green">E45</font></i></a>, <a class="txt" href="lattice8.html#E43:48"><i><font color="Green">E59</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="lattice8.html#E28:48"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice8.html#E39:48"><i><font color="Green">E56</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>19</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E13:48"><i><font color="Green">E43</font></i></a>, <a class="txt" href="lattice8.html#E14:48"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E35:48"><i><font color="Green">E52</font></i></a>, <a class="ref" href="lattice5.html#D7">LATTICE5:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/>


</div>
