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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>;<br/>



<a NAME="E1:26"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="yellow15.html#K2" title="YELLOW15:func.2">Components</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:26_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">Y</font>
 <b>and </b><br/><a NAME="E4:26_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 <b>and </b><br/><a NAME="E5:26_1"/><i><font color="Green" title="E14">A2</font></i>: 
 <a href="yellow15.html#K2" title="YELLOW15:func.2">Components</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">q</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">q</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="yellow15.html#D2" target="_self" title="YELLOW15:def.2">Def2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  =  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  =  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ;<br/>
<a NAME="E6:26_1"/><i><font color="Green" title="E15">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font></span>)</span> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  ) )
 
;<br/>
<b>consider </b><font color="Maroon" title="c5">q</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <b> such that </b><br/><a NAME="E8:26_1"/><i><font color="Green" title="E16">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>
 <b>and </b><br/><a NAME="E9:26_1"/><i><font color="Green" title="E17">A5</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font></span>)</span> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] implies <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] implies <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <b>from </b><i><a class="ref" href="yellow15.html#S1" target="_self" title="YELLOW15:sch.1">YELLOW15:sch 1</a>(<a class="txt" href="yellow15.html#E6:26_1"><i><font color="Green" title="E15">A3</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:26_1_1"/>
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:26_1_1"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span>
 <b>and </b><br/><a NAME="E4:26_1_1"/><i><font color="Green" title="E19">A7</font></i>: 
<span class="p1">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><a NAME="E5:26_1_1"/><i><font color="Green" title="E20">A8</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="yellow15.html#T5" target="_self" title="YELLOW15:th.5">Th5</a></i>;<br/><a NAME="E6:26_1_1"/><b>then </b><i><font color="Green" title="E21">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1_1"/>
( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> or  not <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_1"/><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="yellow15.html#E9:26_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="yellow15.html#E3:26_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="yellow15.html#E6:26_1_1"><i><font color="Green" title="E21">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_1_1_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="yellow15.html#E4:26_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="yellow15.html#E1:26_1_1_1_1_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="yellow15.html#T6" target="_self" title="YELLOW15:th.6">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_2"/><i><font color="Green" title="E22">A11</font></i>: 
not <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="yellow15.html#E9:26_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="yellow15.html#E3:26_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="yellow15.html#E6:26_1_1"><i><font color="Green" title="E21">A9</font></i></a></i>;<br/><a NAME="E3:26_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow15.html#E3:26_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="yellow15.html#E5:26_1_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="yellow15.html#T7" target="_self" title="YELLOW15:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E4:26_1_1_1_1_2"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="yellow15.html#E1:26_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="yellow15.html#E4:26_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="yellow15.html#E1:26_1_1_1_1_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:26_1_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E11:26_1"/><b>then </b><i><font color="Green" title="E18">A12</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow15.html#E1:26_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/>
<a NAME="E12:26_1"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow15.html#K2" title="YELLOW15:func.2">Components</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="yellow15.html#E5:26_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="yellow15.html#E8:26_1"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:26_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="yellow15.html#K2" title="YELLOW15:func.2">Components</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow15.html#E11:26_1"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="yellow15.html#K2" title="YELLOW15:func.2">Components</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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