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

<span class="kw">let </span><font color="Maroon" title="c1">z</font>, <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font>, <font color="Maroon" title="c5">D</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"> ( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> implies ( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:7"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>
 ; <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="c1">z</font> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )</span><br/>

<a NAME="E2:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A2</font></span>: 
(  not <font color="Maroon" title="c4">C</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  not <font color="Maroon" title="c5">D</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T51" title="MCART_1:th.51">MCART_1:51</a></span>;<br/>
<a NAME="E3:7"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
(  not <font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  not <font color="Maroon" title="c3">B</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="mcart_1.html#T51" title="MCART_1:th.51">MCART_1:51</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c7">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c8">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">C</font>, <font color="Maroon" title="c9">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font><span class="kw"> such that </span><br/><a NAME="E5:7"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">d</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="domain_1.html#T10" title="DOMAIN_1:th.10">DOMAIN_1:10</a></span>;<br/>
<a NAME="E6:7"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="recdef_2.html#D6" target="_self" title="RECDEF_2:def.6">Def6</a>, <a class="ref" href="recdef_2.html#D7" target="_self" title="RECDEF_2:def.7">Def7</a></span>;<br/>
<a NAME="E7:7"/>
( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a>, <a class="ref" href="recdef_2.html#D5" target="_self" title="RECDEF_2:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:7"/>
( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E6:7"><span class="lab"><font color="Green" title="E15">A5</font></span></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>
