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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:45_2_1"/>
 <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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="E1:45_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="armstrng.html#M1" target="_self">Element</a> of  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:45_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E4:45_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E5:45_2_1_1"/>
<font color="Maroon">c<sub>4</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="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:45_2_1"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then </b><i><font color="Green">E25</font></i>:  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#D6">RELAT_1:def 6</a></i>
<br/>.= 
 <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<b>thus </b><a NAME="E4:45_2_1"/>
 <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:45_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<a NAME="E5:45_2_1"/>
 <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a><br/>

<b>assume </b><a NAME="E1:45_2_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="armstrng.html#M1" target="_self">Element</a> of  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:45_2_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<b>hence </b><a NAME="E4:45_2_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:45_2_1"/>
 <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V1">reflexive</a>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:45_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>

<a NAME="E7:45_2_1"/>
 <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_2_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E2:45_2_1_4"/><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="armstrng.html#M1" target="_self">Element</a> of  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:45_2_1_4"/><i><font color="Green">E27</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E2:45_2_1_4"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="armstrng.html#M1" target="_self">Element</a> of  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:45_2_1_4"/><i><font color="Green">E28</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E2:45_2_1_4"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E7:45_2_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:45_2_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="armstrng.html#E6:45_2_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E8:45_2_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:45_2_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="armstrng.html#E6:45_2_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="armstrng.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:45_2_1"/>
 <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V4">antisymmetric</a>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:45_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a></i>;<br/>

<a NAME="E9:45_2_1"/>
 <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_2_1_5"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E2:45_2_1_5"/><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="armstrng.html#M1" target="_self">Element</a> of  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:45_2_1_5"/><i><font color="Green">E27</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E2:45_2_1_5"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="armstrng.html#M1" target="_self">Element</a> of  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:45_2_1_5"/><i><font color="Green">E28</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E2:45_2_1_5"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E7:45_2_1_5"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:45_2_1_5"><i><font color="Green">E27</font></i></a>, <a class="txt" href="armstrng.html#E6:45_2_1_5"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:45_2_1_5"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:45_2_1_5"><i><font color="Green">E27</font></i></a>, <a class="txt" href="armstrng.html#E6:45_2_1_5"><i><font color="Green">E28</font></i></a>, <a class="ref" href="armstrng.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E9:45_2_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E7:45_2_1_5"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:45_2_1"/>
 <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V8">(F2)</a>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:45_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>


</div>
