<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">B</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="proofs_1.html#NM2" title="PROOFS_1:NM.2">Rule</a><br/>  for <font color="Olive" title="b2">S</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a>  st (  for <font color="Olive" title="b3">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">S</font> holds <br/><font color="Maroon" title="c1">B</font>,<font color="Olive" title="b1">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Olive" title="b3">a</font> ) holds <br/> ex <font color="Olive" title="b3">S1</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a> st <br/>( <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">S1</font> &amp; <font color="Olive" title="b3">S1</font> is <font color="Maroon" title="c1">B</font>,<font color="Olive" title="b1">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="proofs_1.html#NM2" title="PROOFS_1:NM.2">Rule</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a>  st (  for <font color="Olive" title="b2">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Olive" title="b2">a</font> ) holds <br/> ex <font color="Olive" title="b2">S1</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a> st <br/>( <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">S1</font> &amp; <font color="Olive" title="b2">S1</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> holds <br/><font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Olive" title="b1">a</font> ) implies  ex <font color="Olive" title="b1">S1</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a> st <br/>( <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> &amp; <font color="Olive" title="b1">S1</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:44"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> holds <br/><font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Olive" title="b1">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">S1</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a> st <br/>( <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> &amp; <font color="Olive" title="b1">S1</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">S1</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a> st <br/>( $1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> &amp; <font color="Olive" title="b1">S1</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  );<br/>
<a NAME="E2:44"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c3">S</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E3:44"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="44_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c4">P</font> =  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>;<br/>
<span class="kw">take </span>
 <a href="proofs_1.html#K1" title="PROOFS_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="proofs_1.html#K1" title="PROOFS_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span> &amp;  <a href="proofs_1.html#K1" title="PROOFS_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  )</span><br/>

<a NAME="E1:44_1"/>
 <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V2" title="PROOFS_1:attr.2">-correct</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E2:44_1"/>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="proofs_1.html#K1" title="PROOFS_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span> &amp;  <a href="proofs_1.html#K1" title="PROOFS_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:44"/><span class="lab"><font color="Green" title="E21">A20</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">B1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Olive" title="b2">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B1</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="44_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">B1</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c5">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">B1</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">B1</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="c4">x</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="E1:44_2"/><span class="lab"><font color="Green" title="E22">A21</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">and </span><br/><a NAME="E2:44_2"/>
<font color="Maroon" title="c5">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">and </span><br/><a NAME="E3:44_2"/><span class="lab"><font color="Green" title="E23">A23</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">B1</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">B1</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c4">x</font> as    <a href="proofs_1.html#NM1" title="PROOFS_1:NM.1">Formula</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">S1</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a><span class="kw"> such that </span><br/><a NAME="E6:44_2"/><span class="lab"><font color="Green" title="E24">A24</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S1</font>
 <span class="kw">and </span><br/><a NAME="E7:44_2"/><span class="lab"><font color="Green" title="E25">A25</font></span>: 
<font color="Maroon" title="c7">S1</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E1:44_2"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="proofs_1.html#T7" target="_self" onmouseover="rs('proofs_1/T7')" onmouseout="rh()">Th45</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">S2</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a><span class="kw"> such that </span><br/><a NAME="E9:44_2"/><span class="lab"><font color="Green" title="E26">A26</font></span>: 
<font color="Maroon" title="c5">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">S2</font>
 <span class="kw">and </span><br/><a NAME="E10:44_2"/><span class="lab"><font color="Green" title="E27">A27</font></span>: 
<font color="Maroon" title="c8">S2</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_2"><span class="lab"><font color="Green" title="E23">A23</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">S1</font> <a href="proofs_1.html#K3" title="PROOFS_1:func.3">\/</a> <font color="Maroon" title="c8">S2</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">B1</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="c4">x</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="c7">S1</font> <a href="proofs_1.html#K3" title="PROOFS_1:func.3">\/</a> <font color="Maroon" title="c8">S2</font> &amp; <font color="Maroon" title="c7">S1</font> <a href="proofs_1.html#K3" title="PROOFS_1:func.3">\/</a> <font color="Maroon" title="c8">S2</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  )</span><br/>

<a NAME="E11:44_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</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="c7">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:44_2"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:44_2"/>
<font color="Maroon" title="c5">B1</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="c4">x</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="c7">S1</font> <a href="proofs_1.html#K3" title="PROOFS_1:func.3">\/</a> <font color="Maroon" title="c8">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:44_2"><span class="lab"><font color="Green" title="E26">A26</font></span></a>, <a class="ref" href="xboole_1.html#T13" onmouseover="rs('xboole_1/T13')" onmouseout="rh()">XBOOLE_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">S1</font> <a href="proofs_1.html#K3" title="PROOFS_1:func.3">\/</a> <font color="Maroon" title="c8">S2</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a> </span><br/>

<span class="kw">thus </span><a NAME="E13:44_2"/>
<font color="Maroon" title="c7">S1</font> <a href="proofs_1.html#K3" title="PROOFS_1:func.3">\/</a> <font color="Maroon" title="c8">S2</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:44_2"><span class="lab"><font color="Green" title="E25">A25</font></span></a>, <a class="txt" href="#E10:44_2"><span class="lab"><font color="Green" title="E27">A27</font></span></a>, <a class="ref" href="proofs_1.html#T4" target="_self" onmouseover="rs('proofs_1/T4')" onmouseout="rh()">Th43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E5:44"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">S</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finset_1.html#S2" onmouseover="rs('finset_1/S2')" onmouseout="rh()">FINSET_1:sch 2</a>(<a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E4:44"><span class="lab"><font color="Green" title="E21">A20</font></span></a>);</span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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