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

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font>, <font color="Olive" title="b3">X3</font>, <font color="Olive" title="b4">X4</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b5">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">X2</font>,<font color="Olive" title="b3">X3</font>,<font color="Olive" title="b4">X4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> holds <br/> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  iff for <font color="Olive" title="b6">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X1</font><br/> for <font color="Olive" title="b7">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">X2</font><br/> for <font color="Olive" title="b8">x3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b3">X3</font><br/> for <font color="Olive" title="b9">x4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b4">X4</font>  st <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Olive" title="b6">x1</font>,<font color="Olive" title="b7">x2</font>,<font color="Olive" title="b8">x3</font>,<font color="Olive" title="b9">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> holds <br/><font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">x1</font> )</span><br/><b>let </b><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font>, <font color="Maroon" title="c4">X3</font>, <font color="Maroon" title="c5">X4</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">X3</font>,<font color="Maroon" title="c5">X4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> holds <br/> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  iff for <font color="Olive" title="b2">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X1</font><br/> for <font color="Olive" title="b3">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X2</font><br/> for <font color="Olive" title="b4">x3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font><br/> for <font color="Olive" title="b5">x4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X4</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b4">x3</font>,<font color="Olive" title="b5">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> holds <br/><font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x1</font> )</span><br/><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">X3</font>,<font color="Maroon" title="c5">X4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  iff for <font color="Olive" title="b1">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X1</font><br/> for <font color="Olive" title="b2">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X2</font><br/> for <font color="Olive" title="b3">x3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font><br/> for <font color="Olive" title="b4">x4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X4</font>  st <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b4">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> holds <br/><font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> )</span><br/>











<b>thus </b><a NAME="E1:15"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  implies for <font color="Olive" title="b1">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X1</font><br/> for <font color="Olive" title="b2">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X2</font><br/> for <font color="Olive" title="b3">x3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font><br/> for <font color="Olive" title="b4">x4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X4</font>  st <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b4">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> holds <br/><font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X1</font><br/> for <font color="Olive" title="b2">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X2</font><br/> for <font color="Olive" title="b3">x3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font><br/> for <font color="Olive" title="b4">x4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X4</font>  st <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b4">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> holds <br/><font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> ) implies <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X1</font><br/> for <font color="Olive" title="b2">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X2</font><br/> for <font color="Olive" title="b3">x3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font><br/> for <font color="Olive" title="b4">x4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X4</font>  st <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b4">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> holds <br/><font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font></span><br/>

<b>let </b><font color="Maroon" title="c7">x1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X2</font><br/> for <font color="Olive" title="b2">x3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font><br/> for <font color="Olive" title="b3">x4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X4</font>  st <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font>,<font color="Olive" title="b3">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> holds <br/><font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font></span><br/><b>let </b><font color="Maroon" title="c8">x2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font><br/> for <font color="Olive" title="b2">x4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X4</font>  st <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Olive" title="b1">x3</font>,<font color="Olive" title="b2">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> holds <br/><font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font></span><br/><b>let </b><font color="Maroon" title="c9">x3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X4</font>  st <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Olive" title="b1">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> holds <br/><font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font></span><br/><b>let </b><font color="Maroon" title="c10">x4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X4</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> implies <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> )</span><br/>







<b>assume </b><a NAME="E2:15_1"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font></span><br/>

<a NAME="E3:15_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T60" title="MCART_1:th.60">MCART_1:60</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font>
 <b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E2:15_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="mcart_1.html#T33" title="MCART_1:th.33">MCART_1:33</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>thus </b><a NAME="E2:15"/>
( ( for <font color="Olive" title="b1">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X1</font><br/> for <font color="Olive" title="b2">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X2</font><br/> for <font color="Olive" title="b3">x3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font><br/> for <font color="Olive" title="b4">x4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X4</font>  st <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b4">x4</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> holds <br/><font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> ) implies <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  )
 <b>by </b><i><a class="ref" href="mcart_1.html#T79" title="MCART_1:th.79">MCART_1:79</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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