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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="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</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#K2" title="RECDEF_2:func.2">`2_3</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#K3" title="RECDEF_2:func.3">`3_3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:3"/><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="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</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#K2" title="RECDEF_2:func.2">`2_3</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#K3" title="RECDEF_2:func.3">`3_3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )</span><br/>

<a NAME="E2:3"/><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="E3:3"/><span class="lab"><font color="Green" title="E7">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="#E1:3"><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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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="c6">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="c7">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">C</font><span class="kw"> such that </span><br/><a NAME="E5:3"/><span class="lab"><font color="Green" title="E8">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#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">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="#E1:3"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:3"><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/>
<a NAME="E6:3"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K3" title="RECDEF_2:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="recdef_2.html#D3" target="_self" title="RECDEF_2:def.3">Def3</a></span>;<br/>
<a NAME="E7:3"/>
( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="recdef_2.html#D1" target="_self" title="RECDEF_2:def.1">Def1</a>, <a class="ref" href="recdef_2.html#D2" target="_self" title="RECDEF_2:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:3"/>
( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</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#K2" title="RECDEF_2:func.2">`2_3</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#K3" title="RECDEF_2:func.3">`3_3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E6:3"><span class="lab"><font color="Green" title="E9">A5</font></span></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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