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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_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="E1:24_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</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>  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E2:24_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><a NAME="E3:24_1"/><b>then </b><i><font color="Green">E12</font></i>: 
( <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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="finset_1.html#T14">FINSET_1:14</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font>;<br/><b>thus </b><a NAME="E4:24_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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>thus </b><a NAME="E5:24_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:24_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24"/>
 <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font> is <a href="cohsp_1.html#V1" target="_self">c=directed</a>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T6" target="_self">Th6</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><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="E1:24_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</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>  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E3:24_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>thus </b><a NAME="E4:24_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finsub_1.html#T18">FINSUB_1:18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:24"/>
 <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font> is <a href="cohsp_1.html#V2" target="_self">c=filtered</a>
 <b>by </b><i><a class="ref" href="cohsp_1.html#T8" target="_self">Th8</a></i>;<br/>


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