<?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_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="waybel16.html#M1">CLHomomorphism</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="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:35"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V17">infs-preserving</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 
<b>by </b><i><a class="ref" href="waybel16.html#D1">WAYBEL16:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E4:35"/><i><font color="Green">E25</font></i>: 
 <a href="relset_1.html#K4">dom</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>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:35"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:35"/><i><font color="Green">E27</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E7:35"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E8:35"/><b>then </b>
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="waybel20.html#E4:35"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span> <b>by </b><i><a class="txt" href="waybel20.html#E1:35"><i><font color="Green">E23</font></i></a>, <a class="txt" href="waybel20.html#E5:35"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel20.html#E6:35"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E10:35"/><i><font color="Green">E28</font></i>: 
not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>9</sub></font> is <a href="struct_0.html#V3">empty</a>
 
;<br/>
<a NAME="E11:35"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <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="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E12:35"/><i><font color="Green">E30</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>4</sub></font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D18">YELLOW_0:def 18</a><br/>

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 ;<br/>

<a NAME="E2:35_1"/>
<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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
;<br/>
<a NAME="E3:35_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</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>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:35_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E7:35_1"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E11:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E8:35_1"/>
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> is <a href="waybel_0.html#V17">infs-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel20.html#E2:35"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel20.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E9:35_1"/><b>then </b>
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>
<a NAME="E10:35_1"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <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>11</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:35_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/>
<a NAME="E11:35_1"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E3:35_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E12:35_1"/>
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K6">id</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="waybel20.html#E1:35"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T145">FUNCT_1:145</a></i>;<br/>
<a NAME="E13:35_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K6">id</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="waybel20.html#E11:35_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:35_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E15:35_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="waybel20.html#E13:35_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel20.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E16:35_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:35"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel20.html#E6:35_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E17:35_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</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"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E18:35_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:35_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel20.html#E10:35_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel20.html#E15:35_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E19:35_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span> <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>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E1:35"><i><font color="Green">E23</font></i></a>, <a class="txt" href="waybel20.html#E7:35_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:35"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be  <a href="waybel_0.html#V1">directed</a> <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>4</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D4">WAYBEL_0:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:35_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:35_2"/><i><font color="Green">E32</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 ;<br/>

<a NAME="E3:35_2"/>
<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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
;<br/>
<a NAME="E4:35_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <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 <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span> <b>by </b><i><a class="txt" href="waybel20.html#E10:35"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel20.html#E1:35_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="yellow_2.html#T7">YELLOW_2:7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:35_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E8:35_2"/><i><font color="Green">E35</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E11:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E9:35_2"/>
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel20.html#E2:35"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel20.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E10:35_2"/><b>then </b>
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E11:35_2"/><b>then </b><i><font color="Green">E36</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <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>11</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E2:35_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<a NAME="E12:35_2"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:35_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E13:35_2"/>
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K6">id</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="waybel20.html#E1:35"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T145">FUNCT_1:145</a></i>;<br/>
<a NAME="E14:35_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K6">id</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="waybel20.html#E12:35_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:35_2"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E16:35_2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</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="waybel20.html#E14:35_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel20.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E17:35_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:35"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel20.html#E7:35_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E18:35_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</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"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E19:35_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E8:35_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel20.html#E11:35_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel20.html#E16:35_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E20:35_2"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span> <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>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E1:35"><i><font color="Green">E23</font></i></a>, <a class="txt" href="waybel20.html#E8:35_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:35"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="waybel_5.html#NM3">CLSubFrame</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E10:35"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel20.html#E12:35"><i><font color="Green">E30</font></i></a></i>;<br/>


</div>
