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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> 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="Red">::  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#K3" title="ZFMISC_1:func.3">[:</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></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> implies <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#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K3" title="RECDEF_2:func.3">`3_3</a> </span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:4"/><span class="lab"><font color="Green" title="E5">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#K3" title="ZFMISC_1:func.3">[:</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></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K3" title="RECDEF_2:func.3">`3_3</a> </span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span></span><br/>

<a NAME="E2:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T35" title="MCART_1:th.35">MCART_1:35</a></span>;<br/>
<a NAME="E3:4"/>
( 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="#E1:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="mcart_1.html#T35" title="MCART_1:th.35">MCART_1:35</a></span>;<br/>
<a NAME="E4:4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> ex <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">B</font> ex <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">C</font> st <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#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="domain_1.html#T15" title="DOMAIN_1:th.15">DOMAIN_1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:4"/>
<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#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K3" title="RECDEF_2:func.3">`3_3</a> </span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="recdef_2.html#T1" target="_self" title="RECDEF_2:th.1">Th1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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