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

<b>set </b><font color="Maroon" title="c2">dox</font> =  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c3">dx</font> =  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E1:45_2_1"/>
 <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</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" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:45_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as    <a href="armstrng.html#M1" title="ARMSTRNG:mode.1">Element</a> of  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:45_2_1_1"/>
<font color="Maroon" title="c5">x'</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c5">x'</font>
 
;<br/>
<a NAME="E4:45_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<b>hence </b><a NAME="E5:45_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:45_2_1"/><b>then </b><i><font color="Green" title="E24">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>then </b><i><font color="Green" title="E25">A2</font></i>:  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#D6" title="RELAT_1:def.6">RELAT_1:def 6</a></i>
<br/>.= 
 <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<b>thus </b><a NAME="E4:45_2_1"/>
 <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:45_2_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> &amp;  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> &amp;  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">(F2)</a> )</span><br/>

<a NAME="E5:45_2_1"/>
 <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</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" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:45_2_1_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as    <a href="armstrng.html#M1" title="ARMSTRNG:mode.1">Element</a> of  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:45_2_1_3"/>
<font color="Maroon" title="c5">x'</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c5">x'</font>
 
;<br/>
<b>hence </b><a NAME="E4:45_2_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:45_2_1"/>
 <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:45_2_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> &amp;  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">(F2)</a> )</span><br/>

<a NAME="E7:45_2_1"/>
 <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</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" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> or  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_2_1_4"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> )
 <b>and </b><br/><a NAME="E2:45_2_1_4"/><i><font color="Green" title="E26">A3</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">x'</font>, <font color="Maroon" title="c7">y'</font> being    <a href="armstrng.html#M1" title="ARMSTRNG:mode.1">Element</a> of  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:45_2_1_4"/><i><font color="Green" title="E27">A4</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x'</font>,<font color="Maroon" title="c7">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">x'</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c7">y'</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E2:45_2_1_4"><i><font color="Green" title="E26">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">y''</font>, <font color="Maroon" title="c9">x''</font> being    <a href="armstrng.html#M1" title="ARMSTRNG:mode.1">Element</a> of  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:45_2_1_4"/><i><font color="Green" title="E28">A5</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y''</font>,<font color="Maroon" title="c9">x''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">y''</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c9">x''</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E2:45_2_1_4"><i><font color="Green" title="E26">A3</font></i></a></i>;<br/>
<a NAME="E7:45_2_1_4"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y''</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:45_2_1_4"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="armstrng.html#E6:45_2_1_4"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E8:45_2_1_4"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:45_2_1_4"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="armstrng.html#E6:45_2_1_4"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:45_2_1"/>
 <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:45_2_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">(F2)</a></span><br/>

<a NAME="E9:45_2_1"/>
 <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</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" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> or  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> or  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_2_1_5"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> )
 <b>and </b><br/><a NAME="E2:45_2_1_5"/><i><font color="Green" title="E26">A6</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span><br/>

<b>consider </b><font color="Maroon" title="c7">x'</font>, <font color="Maroon" title="c8">y'</font> being    <a href="armstrng.html#M1" title="ARMSTRNG:mode.1">Element</a> of  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:45_2_1_5"/><i><font color="Green" title="E27">A7</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x'</font>,<font color="Maroon" title="c8">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">x'</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c8">y'</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E2:45_2_1_5"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y''</font>, <font color="Maroon" title="c10">z'</font> being    <a href="armstrng.html#M1" title="ARMSTRNG:mode.1">Element</a> of  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:45_2_1_5"/><i><font color="Green" title="E28">A8</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y''</font>,<font color="Maroon" title="c10">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">y''</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c10">z'</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E2:45_2_1_5"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/>
<a NAME="E7:45_2_1_5"/><i><font color="Green" title="E29">A9</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x'</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y'</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y''</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z'</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:45_2_1_5"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="armstrng.html#E6:45_2_1_5"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:45_2_1_5"/><b>then </b>
<font color="Maroon" title="c7">x'</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c10">z'</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:45_2_1_5"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="armstrng.html#E6:45_2_1_5"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="armstrng.html#T14" target="_self" title="ARMSTRNG:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E9:45_2_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E7:45_2_1_5"><i><font color="Green" title="E29">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:45_2_1"/>
 <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">(F2)</a>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:45_2_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
