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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel_1.html#V6">projection</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V17">infs-preserving</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel_1.html#T46">WAYBEL_1:46</a></i>;<br/>
<a NAME="E3:49"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="yellow_0.html#V8">sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T52">WAYBEL_1:52</a></i>;<br/>
<a NAME="E4:49"/><i><font color="Green">E35</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:49"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_1.html#T54">WAYBEL_1:54</a></i>;<br/>
<a NAME="E5:49"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="txt" href="waybel20.html#E3:49"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_5.html#T28">WAYBEL_5:28</a></i>;<br/>
<a NAME="E6:49"/><i><font color="Green">E37</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font> is <a href="lattice3.html#V3">complete</a>
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:49"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_2.html#T32">YELLOW_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="waybel_1.html#T48">WAYBEL_1:48</a></i>;<br/>
<a NAME="E8:49"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_1.html#V7">closure</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T50">WAYBEL_1:50</a></i>;<br/>
<a NAME="E9:49"/><i><font color="Green">E39</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel15.html#T1">WAYBEL15:1</a></i>;<br/>
<i><font color="Green">E40</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="waybel_1.html#T47">WAYBEL_1:47</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>
;<br/>
<i><font color="Green">E41</font></i>:  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel20.html#E9:49"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel20.html#E10:49"><i><font color="Green">E40</font></i></a>, <a class="ref" href="yellow_0.html#T58">YELLOW_0:58</a></i>
<br/>.= 
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>4</sub></font>

;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ;<br/>
<a NAME="E13:49"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V17">infs-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a><br/>

<b>assume </b><a NAME="E1:49_3"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>6</sub></font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a><br/>

<b>thus </b><a NAME="E2:49_3"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E6:49"><i><font color="Green">E37</font></i></a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E3:49_3"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:49_3"/><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:49_3"/><i><font color="Green">E43</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E7:49_3"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:49"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_0.html#D18">YELLOW_0:def 18</a></i>;<br/>
<a NAME="E8:49_3"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E6:49_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow_0.html#T64">YELLOW_0:64</a></i>;<br/>
<a NAME="E9:49_3"/><i><font color="Green">E45</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E10:49_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:49"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>
<a NAME="E11:49_3"/><i><font color="Green">E47</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E12:49_3"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E13:49_3"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:49"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_0.html#D18">YELLOW_0:def 18</a></i>;<br/>
<a NAME="E14:49_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
;<br/>
<a NAME="E15:49_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E16:49_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>;<br/>
<b>hence </b> <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel20.html#E11:49_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel20.html#E13:49_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="yellow_0.html#T64">YELLOW_0:64</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel20.html#E9:49_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel20.html#E10:49_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel20.html#E5:49_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel20.html#E8:49_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E14:49"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V17">infs-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel20.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E15:49"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel20.html#E8:49"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel_1.html#T58">WAYBEL_1:58</a></i>;<br/>
<a NAME="E16:49"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a><br/>

<b>assume </b><a NAME="E1:49_4"/>
( not <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<b>thus </b><a NAME="E2:49_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E15:49"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:49"/>
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:49"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel20.html#E6:49"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel20.html#E11:49"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel20.html#E14:49"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel_5.html#T33">WAYBEL_5:33</a></i>;<br/>

<b>thus </b><a NAME="E18:49"/>
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 <b>by </b><i><a class="txt" href="waybel20.html#E1:49"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel20.html#E4:49"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel_1.html#T54">WAYBEL_1:54</a></i>;<br/>


</div>
