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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K5">\+\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="finsub_1.html#E1:9"><i><font color="Green">E7</font></i></a></i>;<br/><a NAME="E3:9_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K5">\+\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T98">XBOOLE_1:98</a></i>;<br/><b>hence </b><a NAME="E4:9_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="finsub_1.html#E1:9"><i><font color="Green">E7</font></i></a>, <a class="txt" href="finsub_1.html#E1:9_1"><i><font color="Green">E8</font></i></a></i>;<br/><b>thus </b><a NAME="E5:9_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="finsub_1.html#E1:9"><i><font color="Green">E7</font></i></a>, <a class="txt" href="finsub_1.html#E1:9_1"><i><font color="Green">E8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="finsub_1.html#V4" target="_self">preBoolean</a>
 <b>by </b><i><a class="ref" href="finsub_1.html#T10" target="_self">Th10</a></i>;<br/>


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