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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>1</sub></font> is <a href="xboole_0.html#V1">zero</a> or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:16_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E2:16_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E3:16_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or ( not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">zero</a> &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) or <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">zero</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E3:16_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>]
 <b>by </b><i><a class="txt" href="chain_1.html#E1:16_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_1_2"/><i><font color="Green">E13</font></i>: 
( not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">zero</a> &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:16_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>hence </b><a NAME="E3:16_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>]
 <b>by </b><i><a class="txt" href="chain_1.html#E2:16"><i><font color="Green">E8</font></i></a>, <a class="txt" href="chain_1.html#E1:16_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="chain_1.html#E1:16_1_1_1_2"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_3"><b>suppose </b></a><a NAME="E1:16_1_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">zero</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>]
 <b>by </b><i><a class="txt" href="chain_1.html#E1:16"><i><font color="Green">E7</font></i></a>, <a class="txt" href="chain_1.html#E1:16_1_1"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:16_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 
<b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(<a class="txt" href="chain_1.html#E1:16_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="chain_1.html#E2:16_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="chain_1.html#E3:16_1"><i><font color="Green">E11</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:16_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 ;<br/>


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