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

<span class="kw">let </span><font color="Maroon" title="c1">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">Si</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">ASeq</font> being  <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b1">Si</font> holds   <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b2">ASeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Si</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Si</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <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"> for <font color="Olive" title="b1">ASeq</font> being  <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font> holds   <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b1">ASeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">ASeq</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c3">ASeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font></span><br/>





<a NAME="E1:61"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font> holds <br/>for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <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"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font> implies for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</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> holds  <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">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"> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font></span><br/>

<a NAME="E2:61_1"/>
<font color="Maroon" title="c4">A1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></span>;<br/>
<a NAME="E3:61_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="prob_1.html#D1" target="_self" title="PROB_1:def.1">Def1</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:61_1"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D4" target="_self" title="PROB_1:def.4">Def4</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="E2:61"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font> holds <br/> <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Olive" title="b1">A1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <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"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font> implies  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font></span><br/>

<a NAME="E2:61_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E20">A1</font></span></a></span>;<br/>
<a NAME="E3:61_2"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></span>;<br/>
<a NAME="E4:61_2"/><span class="kw">then </span>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D8" target="_self" title="PROB_1:def.8">Def8</a></span>;<br/>
<a NAME="E5:61_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p3">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <span class="p4">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span></span>)</span></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>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D1" target="_self" title="PROB_1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:61_2"/>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</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>
<a NAME="E3:61"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font> holds <br/> <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <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"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font> implies  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font></span><br/>

<a NAME="E2:61_3"/><span class="kw">then </span>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c4">A1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:61_3"/>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</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>
<a NAME="E4:61"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ASeq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:61"/>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c3">ASeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E22">A4</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>
