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

<span class="kw">set </span><font color="Maroon" title="c5">a</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:55_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T22" onmouseover="rs('trees_1/T22')" onmouseout="rh()">TREES_1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:55_1_1"/>
 not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</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">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<a NAME="E3:55_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p5">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K2" title="LTLAXIO3:func.2">**</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p5">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K2" title="LTLAXIO3:func.2">**</a></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p5">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K2" title="LTLAXIO3:func.2">**</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p5">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K2" title="LTLAXIO3:func.2">**</a></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:55_1_1_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p5">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K2" title="LTLAXIO3:func.2">**</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p5">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K2" title="LTLAXIO3:func.2">**</a></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c6">x</font> as   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a> ;<br/>
<a NAME="E3:55_1_1_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c7">x1</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="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T27" target="_self" onmouseover="rs('ltlaxio4/T27')" onmouseout="rh()">Th27</a>, <a class="txt" href="#E1:55_1_1_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>;<br/>
<a NAME="E4:55_1_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c7">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E5:55_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x1</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a>  is   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="txt" href="#E3:55_1_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E6:55_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p3">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K2" title="LTLAXIO3:func.2">**</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#D2" onmouseover="rs('ltlaxio3/D2')" onmouseout="rh()">LTLAXIO3:def 2</a></span>;<br/>
<a NAME="E7:55_1_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c7">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E8:55_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x1</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a>  is   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="txt" href="#E3:55_1_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E9:55_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p3">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K2" title="LTLAXIO3:func.2">**</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#D2" onmouseover="rs('ltlaxio3/D2')" onmouseout="rh()">LTLAXIO3:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E11:55_1_1_1"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a>  &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a>  )
 <span class="kw">and </span><br/><a NAME="E12:55_1_1_1"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c11">z</font> = <font color="Maroon" title="c9">z</font> as   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:55_1_1_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="ltlaxio3.html#D2" onmouseover="rs('ltlaxio3/D2')" onmouseout="rh()">LTLAXIO3:def 2</a></span>;<br/>
<span class="kw">thus </span><a NAME="E14:55_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p5">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K2" title="LTLAXIO3:func.2">**</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="ltlaxio3.html#K7" title="LTLAXIO3:func.7">Subt</a> <span class="p5">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K2" title="LTLAXIO3:func.2">**</a></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:55_1_1_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E9:55_1_1_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a>, <a class="txt" href="#E12:55_1_1_1"><span class="lab"><font color="Green" title="E35">A6</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:55_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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>
