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<b>let </b><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</font>, <font color="Maroon" title="c3">X3</font>, <font color="Maroon" title="c4">X4</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"> for <font color="Olive" title="b1">A1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X1</font><br/> for <font color="Olive" title="b2">A2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X2</font><br/> for <font color="Olive" title="b3">A3</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X3</font><br/> for <font color="Olive" title="b4">A4</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">X4</font><br/> for <font color="Olive" title="b5">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b5">x</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="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font>,<font color="Olive" title="b3">A3</font>,<font color="Olive" title="b4">A4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> holds <br/>( <font color="Olive" title="b5">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A1</font> &amp; <font color="Olive" title="b5">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A2</font> &amp; <font color="Olive" title="b5">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">A3</font> &amp; <font color="Olive" title="b5">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">A4</font> )</span><br/><b>let </b><font color="Maroon" title="c5">A1</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X2</font><br/> for <font color="Olive" title="b2">A3</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X3</font><br/> for <font color="Olive" title="b3">A4</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">X4</font><br/> for <font color="Olive" title="b4">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b4">x</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="c5">A1</font>,<font color="Olive" title="b1">A2</font>,<font color="Olive" title="b2">A3</font>,<font color="Olive" title="b3">A4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> holds <br/>( <font color="Olive" title="b4">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Olive" title="b4">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A2</font> &amp; <font color="Olive" title="b4">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A3</font> &amp; <font color="Olive" title="b4">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">A4</font> )</span><br/><b>let </b><font color="Maroon" title="c6">A2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A3</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X3</font><br/> for <font color="Olive" title="b2">A4</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">X4</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b3">x</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="c5">A1</font>,<font color="Maroon" title="c6">A2</font>,<font color="Olive" title="b1">A3</font>,<font color="Olive" title="b2">A4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> holds <br/>( <font color="Olive" title="b3">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Olive" title="b3">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A2</font> &amp; <font color="Olive" title="b3">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A3</font> &amp; <font color="Olive" title="b3">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A4</font> )</span><br/><b>let </b><font color="Maroon" title="c7">A3</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A4</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">X4</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b2">x</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="c5">A1</font>,<font color="Maroon" title="c6">A2</font>,<font color="Maroon" title="c7">A3</font>,<font color="Olive" title="b1">A4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> holds <br/>( <font color="Olive" title="b2">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Olive" title="b2">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A2</font> &amp; <font color="Olive" title="b2">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A3</font> &amp; <font color="Olive" title="b2">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A4</font> )</span><br/><b>let </b><font color="Maroon" title="c8">A4</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">X4</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b1">x</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="c5">A1</font>,<font color="Maroon" title="c6">A2</font>,<font color="Maroon" title="c7">A3</font>,<font color="Maroon" title="c8">A4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> holds <br/>( <font color="Olive" title="b1">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Olive" title="b1">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A2</font> &amp; <font color="Olive" title="b1">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A3</font> &amp; <font color="Olive" title="b1">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A4</font> )</span><br/><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</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="c9">x</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="c5">A1</font>,<font color="Maroon" title="c6">A2</font>,<font color="Maroon" title="c7">A3</font>,<font color="Maroon" title="c8">A4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> implies ( <font color="Maroon" title="c9">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Maroon" title="c9">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A2</font> &amp; <font color="Maroon" title="c9">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A3</font> &amp; <font color="Maroon" title="c9">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A4</font> ) )</span><br/>

















<b>assume </b><a NAME="E1:83"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c9">x</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="c5">A1</font>,<font color="Maroon" title="c6">A2</font>,<font color="Maroon" title="c7">A3</font>,<font color="Maroon" title="c8">A4</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="c9">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Maroon" title="c9">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A2</font> &amp; <font color="Maroon" title="c9">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A3</font> &amp; <font color="Maroon" title="c9">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A4</font> )</span><br/>

<a NAME="E2:83"/><b>then </b><i><font color="Green" title="E48">A2</font></i>: 
( <font color="Maroon" title="c5">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">A2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c7">A3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">A4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T55" target="_self" title="MCART_1:th.55">Th55</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c5">A1</font>,<font color="Maroon" title="c6">A2</font>,<font color="Maroon" title="c7">A3</font>,<font color="Maroon" title="c8">A4</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> <b>by </b><i><a class="txt" href="mcart_1.html#E1:83"><i><font color="Green" title="E47">A1</font></i></a></i>;<br/>
<a NAME="E4:83"/>
( <font color="Maroon" title="c10">y</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Maroon" title="c10">y</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A2</font> &amp; <font color="Maroon" title="c10">y</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A3</font> &amp; <font color="Maroon" title="c10">y</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A4</font> )
 
<b>by </b><i><a class="txt" href="mcart_1.html#E2:83"><i><font color="Green" title="E48">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:83"/>
( <font color="Maroon" title="c9">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Maroon" title="c9">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A2</font> &amp; <font color="Maroon" title="c9">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A3</font> &amp; <font color="Maroon" title="c9">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A4</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T86" target="_self" title="MCART_1:th.86">Th86</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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