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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="waybel_1.html#V6">projection</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<a NAME="E1:55"/>
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="yellow_2.html#K1">Image</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> as  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</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/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="ref" href="yellow_2.html#T37">YELLOW_2:37</a></i>;<br/>
<a NAME="E4:55"/><i><font color="Green">E40</font></i>: 
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:55"/>
<font color="Maroon">c<sub>5</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow16.html#T41">YELLOW16:41</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> as  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel27.html#E4:55"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel15.html#T1">WAYBEL15:1</a></i>;<br/>
<a NAME="E7:55"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:55_1"/>
the <a href="struct_0.html#U1">carrier</a> of <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 <font color="Maroon">c<sub>4</sub></font>
  <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="55_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</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="E1:55_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:55_1_1"/><i><font color="Green">E41</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</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="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E4:55_1_1"/><i><font color="Green">E42</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:55_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</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="txt" href="waybel27.html#E3:55_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><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/>
<a NAME="E7:55_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <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="E1:55_1_1_1"/><i><font color="Green">E44</font></i>: 
( not <font color="Maroon">c<sub>10</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<b>thus </b><a NAME="E2:55_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>10</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="waybel27.html#E4:55_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel27.html#E1:55_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="relat_1.html#T152">RELAT_1:152</a>, <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/>
<a NAME="E3:55_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="waybel27.html#E1:55_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E4:55_1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel27.html#E1:55_1_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<b>thus </b><a NAME="E5:55_1_1_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E6:55_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T37">YELLOW_2:37</a></i>;<br/>
<b>hence </b><a NAME="E7:55_1_1_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel27.html#E4:55_1_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel27.html#E5:55_1_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_0.html#T7">WAYBEL_0:7</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E8:55_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>9</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="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E9:55_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E7:55_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel27.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="waybel27.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>
;<br/>
<a NAME="E11:55_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel27.html#E8:55_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel27.html#E9:55_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:55_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</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> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>hence </b><a NAME="E13:55_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>7</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="E2:55_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:55_1"/><b>then </b>
<font color="Maroon">c<sub>7</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> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E4:55_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:55_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</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>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel27.html#E6:55_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel27.html#D4" target="_self">Def4</a></i>;<br/>
<i><font color="Green">E42</font></i>: <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E6:55_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel27.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="waybel15.html#T13">WAYBEL15:13</a></i>;<br/>
<a NAME="E10:55_1"/><i><font color="Green">E43</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:55_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</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> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</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/>
<a NAME="E12:55_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E13:55_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel27.html#E8:55_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="waybel27.html#E10:55_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E15:55_1"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <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="E1:55_1_3"/><i><font color="Green">E44</font></i>: 
( not <font color="Maroon">c<sub>12</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>12</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<b>thus </b><a NAME="E2:55_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>12</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_3_1"/><i><font color="Green">E45</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>12</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel27.html#E10:55_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel27.html#E1:55_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="relat_1.html#T152">RELAT_1:152</a>, <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/>
<a NAME="E3:55_1_3_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="waybel27.html#E1:55_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E4:55_1_3_1"/><b>then </b><i><font color="Green">E47</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel27.html#E1:55_1_3_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<b>thus </b><a NAME="E5:55_1_3_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E6:55_1_3_1"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel27.html#T12" target="_self">Th12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>12</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="waybel27.html#E10:55_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel27.html#E1:55_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="relat_1.html#T152">RELAT_1:152</a>, <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/>
<a NAME="E8:55_1_3_1"/>
<font color="Maroon">c<sub>5</sub></font> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T37">YELLOW_2:37</a></i>;<br/>
<a NAME="E9:55_1_3_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="waybel27.html#E4:55_1_3_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_0.html#T7">WAYBEL_0:7</a></i>;<br/>
<b>hence </b><a NAME="E10:55_1_3_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel27.html#E1:55_1_3_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel27.html#E3:55_1_3_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:55_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:55"/>
 <a href="yellow_2.html#K1">Image</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T58">YELLOW_0:58</a></i>;<br/>


</div>
