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<b>let </b><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>, <font color="Maroon" title="c6">E</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"> ( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_2.html#K2" title="MCART_2:func.2">[:</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>,<font color="Maroon" title="c6">E</font></span><a href="mcart_2.html#K2" title="MCART_2:func.2">:]</a></span> implies ( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K8" title="RECDEF_2:func.8">`1_5</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#K9" title="RECDEF_2:func.9">`2_5</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#K10" title="RECDEF_2:func.10">`3_5</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#K11" title="RECDEF_2:func.11">`4_5</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K12" title="RECDEF_2:func.12">`5_5</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> ) )</span><br/>











<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_2.html#K2" title="MCART_2:func.2">[:</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>,<font color="Maroon" title="c6">E</font></span><a href="mcart_2.html#K2" title="MCART_2:func.2">:]</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">z</font> <a href="recdef_2.html#K8" title="RECDEF_2:func.8">`1_5</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#K9" title="RECDEF_2:func.9">`2_5</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#K10" title="RECDEF_2:func.10">`3_5</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#K11" title="RECDEF_2:func.11">`4_5</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K12" title="RECDEF_2:func.12">`5_5</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> )</span><br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
( 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> &amp; 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> &amp; not <font color="Maroon" title="c6">E</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="ref" href="mcart_2.html#T13" title="MCART_2:th.13">MCART_2:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">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="c8">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="c9">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="c10">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font>, <font color="Maroon" title="c11">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_2.html#K1" title="MCART_2:func.1">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>,<font color="Maroon" title="c11">e</font></span><a href="mcart_2.html#K1" title="MCART_2:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:11"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="mcart_2.html#T17" title="MCART_2:th.17">MCART_2:17</a></i>;<br/>
<a NAME="E5:11"/>
( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K8" title="RECDEF_2:func.8">`1_5</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K9" title="RECDEF_2:func.9">`2_5</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K10" title="RECDEF_2:func.10">`3_5</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K11" title="RECDEF_2:func.11">`4_5</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K12" title="RECDEF_2:func.12">`5_5</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e</font> )
 
<b>by </b><i><a class="txt" href="recdef_2.html#E4:11"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="recdef_2.html#D8" target="_self" title="RECDEF_2:def.8">Def8</a>, <a class="ref" href="recdef_2.html#D9" target="_self" title="RECDEF_2:def.9">Def9</a>, <a class="ref" href="recdef_2.html#D10" target="_self" title="RECDEF_2:def.10">Def10</a>, <a class="ref" href="recdef_2.html#D11" target="_self" title="RECDEF_2:def.11">Def11</a>, <a class="ref" href="recdef_2.html#D12" target="_self" title="RECDEF_2:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E6:11"/>
( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K8" title="RECDEF_2:func.8">`1_5</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#K9" title="RECDEF_2:func.9">`2_5</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#K10" title="RECDEF_2:func.10">`3_5</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#K11" title="RECDEF_2:func.11">`4_5</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K12" title="RECDEF_2:func.12">`5_5</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> )
 <b>by </b><i><a class="txt" href="recdef_2.html#E2:11"><i><font color="Green" title="E17">A2</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/>


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