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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">P</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Olive" title="b2">Q</font> &amp; <font color="Olive" title="b2">Q</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Olive" title="b1">P</font> holds <br/><font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c2">p</font> implies <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:38"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">and </span><br/><a NAME="E2:38"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font></span><br/>

<a NAME="E3:38"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">p</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="armstrng.html#D9" target="_self" title="ARMSTRNG:def.9">Def10</a></span>;<br/>
<a NAME="E4:38"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">q</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="armstrng.html#D9" target="_self" title="ARMSTRNG:def.9">Def10</a></span>;<br/>
<a NAME="E5:38"/>
<font color="Maroon" title="c3">q</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">p</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="armstrng.html#D9" target="_self" title="ARMSTRNG:def.9">Def10</a></span>;<br/>
<a NAME="E6:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E7:38"/>
<font color="Maroon" title="c2">p</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">q</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="armstrng.html#D9" target="_self" title="ARMSTRNG:def.9">Def10</a></span>;<br/>
<a NAME="E8:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E9:38"/>
<font color="Maroon" title="c2">p</font> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T22" title="MCART_1:th.22">MCART_1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:38"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E6:38"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="mcart_1.html#T22" title="MCART_1:th.22">MCART_1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
