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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b1">X</font> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) ) implies <font color="Maroon" title="c1">A</font> is <a href="finsub_1.html#V4" title="FINSUB_1:attr.4">preBoolean</a> )</span><br/>

<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b1">X</font> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">A</font> is <a href="finsub_1.html#V4" title="FINSUB_1:attr.4">preBoolean</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</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="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies ( <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) )</span><br/><b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )</span><br/><b>then reconsider </b><font color="Maroon" title="c4">Z</font> = <font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/><a NAME="E3:9_1"/>
<font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <font color="Maroon" title="c4">Z</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T98" title="XBOOLE_1:th.98">XBOOLE_1:98</a></i>;<br/><b>hence </b><a NAME="E4:9_1"/>
<font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E1:9_1"><i><font color="Green" title="E8">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font></span><br/><b>thus </b><a NAME="E5:9_1"/>
<font color="Maroon" title="c2">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E1:9_1"><i><font color="Green" title="E8">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/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
<font color="Maroon" title="c1">A</font> is <a href="finsub_1.html#V4" title="FINSUB_1:attr.4">preBoolean</a>
 <b>by </b><i><a class="ref" href="finsub_1.html#T10" target="_self" title="FINSUB_1:th.10">Th10</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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