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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="closure3.html#R1" target="_self">is_finer_than</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="closure3.html#R1" target="_self">is_finer_than</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="closure3.html#D1" target="_self">CLOSURE3:def 1</a><br/>

<b>assume </b><a NAME="E3:6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:6"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:6"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E1:6"><i><font color="Green">E6</font></i></a>, <a class="ref" href="closure3.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:6"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E9:6"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E2:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="closure3.html#E5:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="closure3.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E10:6"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="closure3.html#E6:6"><i><font color="Green">E9</font></i></a>, <a class="txt" href="closure3.html#E8:6"><i><font color="Green">E10</font></i></a>, <a class="txt" href="closure3.html#E9:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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