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

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">x</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"> for <font color="Olive" title="b1">F1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">F1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K10" title="PROB_3:func.10">Intersection</a> <font color="Olive" title="b1">F1</font> iff for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">F1</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b2">k</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</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" title="c3">F1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K10" title="PROB_3:func.10">Intersection</a> <font color="Maroon" title="c3">F1</font> iff for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">k</font> ) )</span><br/>





<a NAME="E1:50"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> implies <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_1"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</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" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font></span><br/>

<a NAME="E2:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T55" target="_self" title="PROB_3:th.55">Th55</a></span>;<br/>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D5" title="SUBSET_1:def.5">SUBSET_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="prob_3.html#D8" target="_self" title="PROB_3:def.8">Def8</a></span>
<br/>.= 
<font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font>

;<br/>
<span class="kw">hence </span><a NAME="E4:50_1"/>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font>
 ; <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>
<span class="kw">assume </span><a NAME="E2:50"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c3">F1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K10" title="PROB_3:func.10">Intersection</a> <font color="Maroon" title="c3">F1</font> iff for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">k</font> )</span><br/>

<a NAME="E3:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></span>;<br/>
<a NAME="E4:50"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
( ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/> not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font> ) ) iff for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font> ) implies ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/> not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font> ) ) )</span>
<div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_2_1"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E2:50_2_1"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/> not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font> )</span><br/><span class="kw">assume </span><a NAME="E3:50_2_1"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</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" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font></span><br/><a NAME="E4:50_2_1"/>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_2_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:50_2_1"><span class="lab"><font color="Green" title="E50">A8</font></span></a></span>;<br/><a NAME="E5:50_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2_1"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:50_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:50_2_1"><span class="lab"><font color="Green" title="E50">A8</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>

<span class="kw">assume </span><a NAME="E2:50_2"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</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" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/> not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font> ) )</span><br/>

<div><span class="lab"><font color="Green" title="E49">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> implies ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:50_2_2"/><span class="lab"><font color="Green" title="E50">A11</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</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" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font> )</span><br/><a NAME="E2:50_2_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_2"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E1:50_2_2"><span class="lab"><font color="Green" title="E50">A11</font></span></a></span>;<br/><a NAME="E3:50_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E1:50_2_2"><span class="lab"><font color="Green" title="E50">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:50_2_2"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c4">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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="E4:50_2"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:50_2"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/> not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E49">A10</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>
<a NAME="E5:50"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T55" target="_self" title="PROB_3:th.55">Th55</a></span>;<br/>
<a NAME="E6:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A12</font></span>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> iff ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/> not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T54" target="_self" title="PROB_3:th.54">Th54</a></span>;<br/>
<a NAME="E7:50"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  iff <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K9" title="PROB_3:func.9">Complement</a> <font color="Maroon" title="c3">F1</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D5" title="SUBSET_1:def.5">SUBSET_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:50"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K10" title="PROB_3:func.10">Intersection</a> <font color="Maroon" title="c3">F1</font> iff for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F1</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:50"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="prob_3.html#D9" target="_self" title="PROB_3:def.9">Def9</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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>
