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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> 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="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</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></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> implies ( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</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#K5" title="RECDEF_2:func.5">`2_4</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#K6" title="RECDEF_2:func.6">`3_4</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#K7" title="RECDEF_2:func.7">`4_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> ) )</span><br/>









<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E10">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="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</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></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">z</font> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</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#K5" title="RECDEF_2:func.5">`2_4</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#K6" title="RECDEF_2:func.6">`3_4</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#K7" title="RECDEF_2:func.7">`4_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )</span><br/>

<a NAME="E2:7"/><b>then </b><i><font color="Green" title="E11">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> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T55" title="MCART_1:th.55">MCART_1:55</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">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="c7">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="c8">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="c9">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font><b> such that </b><br/><a NAME="E4:7"/><i><font color="Green" title="E12">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_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">d</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:7"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="domain_1.html#T31" title="DOMAIN_1:th.31">DOMAIN_1:31</a></i>;<br/>
<a NAME="E5:7"/>
( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> )
 
<b>by </b><i><a class="txt" href="recdef_2.html#E4:7"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a>, <a class="ref" href="recdef_2.html#D5" target="_self" title="RECDEF_2:def.5">Def5</a>, <a class="ref" href="recdef_2.html#D6" target="_self" title="RECDEF_2:def.6">Def6</a>, <a class="ref" href="recdef_2.html#D7" target="_self" title="RECDEF_2:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E6:7"/>
( <font color="Maroon" title="c1">z</font> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</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#K5" title="RECDEF_2:func.5">`2_4</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#K6" title="RECDEF_2:func.6">`3_4</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#K7" title="RECDEF_2:func.7">`4_4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )
 <b>by </b><i><a class="txt" href="recdef_2.html#E2:7"><i><font color="Green" title="E11">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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