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<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>, <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:()"><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="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/>











<span class="kw">assume </span><a NAME="E1:11"/><span class="lab"><font color="Green" title="E17">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="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:()"><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#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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">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_2.html#T11" title="MCART_2:th.11">MCART_2:11</a></span>;<br/>
<a NAME="E3:11"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 not <font color="Maroon" title="c6">E</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="#E1:11"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="mcart_2.html#T11" title="MCART_2:th.11">MCART_2:11</a></span>;<br/>
<a NAME="E4:11"/><span class="lab"><font color="Green" title="E20">A4</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="#E1:11"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="mcart_2.html#T11" title="MCART_2:th.11">MCART_2:11</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:11"/><span class="lab"><font color="Green" title="E21">A5</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_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="mcart_2.html#T15" title="MCART_2:th.15">MCART_2:15</a></span>;<br/>
<a NAME="E7:11"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="recdef_2.html#D12" target="_self" title="RECDEF_2:def.12">Def12</a></span>;<br/>
<a NAME="E8:11"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11"><span class="lab"><font color="Green" title="E21">A5</font></span></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></span>;<br/>
<a NAME="E9:11"/>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11"><span class="lab"><font color="Green" title="E21">A5</font></span></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></span>;<br/>
<span class="kw">hence </span><a NAME="E10:11"/>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E8:11"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:11"><span class="lab"><font color="Green" title="E22">A6</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/>


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