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<div class="add">

<a NAME="E1:70_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<font color="Maroon">F<sub>4</sub></font>() is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E2:70_1"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E49">A1</font></span></a></span>;<br/>
<a NAME="E3:70_1"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">C1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon" title="c2">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">C1</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70_1_1"/><span class="lab"><font color="Green" title="E55">A7</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">and </span><br/><a NAME="E2:70_1_1"/><span class="lab"><font color="Green" title="E56">A8</font></span>: 
<font color="Maroon" title="c2">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">and </span><br/><a NAME="E3:70_1_1"/><span class="lab"><font color="Green" title="E57">A9</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">C1</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">A9</font> = <font color="Maroon" title="c1">A</font> as   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">C19</font> = <font color="Maroon" title="c2">C1</font> as   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_1_1"/>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font> or  not <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/><div class="add"><a NAME="E2:70_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><span class="kw">hence </span><a NAME="E3:70_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A10</font></span>: 
not <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">A9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c5">A9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">assume </span><a NAME="E1:70_1_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A11</font></span>: 
<font color="Maroon" title="c5">A9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:70_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A12</font></span>: 
<font color="Maroon" title="c5">A9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:70_1_1_1_2_1"/>
<font color="Maroon" title="c5">A9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_1_2_1"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:70_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_1_2"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="txt" href="#E2:70_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A12</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:70_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">C1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A13</font></span>: <font color="Maroon" title="c4">C19</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">A9</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T101" title="XBOOLE_1:th.101">XBOOLE_1:101</a></span>
<br/>.= 
<font color="Maroon" title="c2">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>

;<br/><a NAME="E5:70_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A14</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">A9</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a></span>;<br/><a NAME="E6:70_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><span class="kw">hence </span><a NAME="E7:70_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="txt" href="#E4:70_1_1_1_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E5:70_1_1_1_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E4:70_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>()]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="#E1:70_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E3:70_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
