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<div class="add">

<b>let </b><font color="Maroon" title="c1">E</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c1">E</font>;<br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>] <b>means </b>( <span class="p1">(<span class="default"><span class="p2">(<span class="default">$2 <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default">$2 <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default">$2 <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default">$2 <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> $2 <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default">$2 <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> $2 <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default">$2 <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> $2 <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> ) <b>-&gt; </b>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c1">E</font> =  <a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,$1;<br/>
<a NAME="E2:39"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>  ex <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">c</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c4">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:39_1"/><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c4">x0</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x0</font> = <font color="Maroon" title="c4">x0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="txt" href="abian.html#E2:39_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being   <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>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Olive" title="b1">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Olive" title="b2">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> );<br/>
<b>set </b><font color="Maroon" title="c6">c1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> ;<br/>
<a NAME="E4:39_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">c</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon" title="c7">c1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being   <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>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; not <font color="Olive" title="b1">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Olive" title="b2">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> );<br/>
<b>set </b><font color="Maroon" title="c8">c2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> ;<br/>
<a NAME="E6:39_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">c</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon" title="c9">c2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">c3</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1"/>
( <font color="Maroon" title="c7">c1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">c2</font> or <font color="Maroon" title="c7">c1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">c2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c7">c1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">c2</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">b</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c7">c1</font>,<font color="Maroon" title="c9">c2</font>,<font color="Maroon" title="c10">c3</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/><a NAME="E2:39_1_1_1"/><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c2</font> &amp; <font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c3</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T47" title="MCART_1:th.47">MCART_1:47</a></i>;<br/><b>thus </b><a NAME="E3:39_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_1_1_1"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_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="E1:39_1_1_1_1_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c2</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c3</font> )
 </span><b>by </b><i><a class="txt" href="abian.html#E2:39_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="abian.html#E1:39_1_1_1_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">xx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font><b> such that </b><br/><a NAME="E3:39_1_1_1_1_1_1_1"/><i><font color="Green" title="E19">A7</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xx</font> &amp;  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being   <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>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Olive" title="b1">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Olive" title="b2">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) )
 ;<br/><b>thus </b><a NAME="E4:39_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_1_1_1_1_1"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">xx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font><b> such that </b><br/><a NAME="E3:39_1_1_1_1_1_1_2"/><i><font color="Green" title="E19">A8</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xx</font> &amp;  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being   <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>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; not <font color="Olive" title="b1">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Olive" title="b2">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) )
 ;<br/><b>thus </b><a NAME="E4:39_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_1_1_1_1_2"><i><font color="Green" title="E19">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_1_1_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c3</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_1_1_1_1_1_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c12">x</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="E2:39_1_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">xc</font> = <font color="Maroon" title="c12">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> ;<br/>
<a NAME="E4:39_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">xc</font>,<font color="Maroon" title="c5">x0</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="abian.html#E2:39_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">k</font>, <font color="Maroon" title="c15">l</font> being    <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> <b> such that </b><br/><a NAME="E6:39_1_1_1_1"/><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c14">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">xc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font>
 <b>by </b><i><a class="ref" href="abian.html#D7" target="_self" title="ABIAN:def.7">Def7</a></i>;<br/>
<a NAME="E7:39_1_1_1_1"/><i><font color="Green" title="E19">A10</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c14">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">l</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1_2"/>
( <font color="Maroon" title="c14">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c14">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2_1"/><i><font color="Green" title="E20">A11</font></i>: 
<font color="Maroon" title="c14">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">k</font> = <font color="Maroon" title="c14">k</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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>  ;<br/><b>thus </b><a NAME="E3:39_1_1_1_1_2_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c15">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c15">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c15">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font>
 <b>by </b><i><a class="txt" href="abian.html#E6:39_1_1_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="abian.html#E1:39_1_1_1_1_2_1"><i><font color="Green" title="E20">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:39_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E2:39_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2_1_1_1_2"/>
not <font color="Maroon" title="c15">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c17">l</font> = <font color="Maroon" title="c15">l</font> as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  ;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">xc</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xc</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E6:39_1_1_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="abian.html#E7:39_1_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="txt" href="abian.html#E7:39_1_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
;<br/><a NAME="E4:39_1_1_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c2</font>
 ;<br/><b>hence </b><a NAME="E5:39_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E2:39_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2_2"/><i><font color="Green" title="E20">A12</font></i>: 
not <font color="Maroon" title="c14">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">k</font> = <font color="Maroon" title="c14">k</font> as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  ;<br/><b>thus </b><a NAME="E3:39_1_1_1_1_2_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1_2_2_1_1"/>
( <font color="Maroon" title="c15">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c15">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c15">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c2</font>
 <b>by </b><i><a class="txt" href="abian.html#E6:39_1_1_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="abian.html#E1:39_1_1_1_1_2_2"><i><font color="Green" title="E20">A12</font></i></a></i>;<br/><b>hence </b><a NAME="E3:39_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E2:39_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2_2_1_1_2"/>
not <font color="Maroon" title="c15">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c17">l</font> = <font color="Maroon" title="c15">l</font> as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  ;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">xc</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xc</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E6:39_1_1_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="abian.html#E7:39_1_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="txt" href="abian.html#E7:39_1_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
;<br/><a NAME="E4:39_1_1_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font>
 ;<br/><b>hence </b><a NAME="E5:39_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E2:39_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E4:39_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c7">c1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">c2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">y</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="E1:39_1_1_1_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c7">c1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1_2_1"/><i><font color="Green" title="E18">A13</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">xx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font><b> such that </b><br/><a NAME="E5:39_1_1_1_2_1"/><i><font color="Green" title="E19">A14</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xx</font> &amp;  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being   <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>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Olive" title="b1">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Olive" title="b2">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) )
 <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_1_2_1"><i><font color="Green" title="E18">A13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">yc</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_1_2_1"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="abian.html#E5:39_1_1_1_2_1"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="abian.html#T6" target="_self" title="ABIAN:th.6">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">k</font>, <font color="Maroon" title="c17">l</font> being    <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> <b> such that </b><br/><a NAME="E8:39_1_1_1_2_1"/><i><font color="Green" title="E20">A15</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Maroon" title="c16">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c17">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 <b>by </b><i><a class="txt" href="abian.html#E5:39_1_1_1_2_1"><i><font color="Green" title="E19">A14</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c16">k</font>, <font color="Maroon" title="c19">l</font> = <font color="Maroon" title="c17">l</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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>  <b>by </b><i><a class="txt" href="abian.html#E8:39_1_1_1_2_1"><i><font color="Green" title="E20">A15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">k1</font> = <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c21">l1</font> = <font color="Maroon" title="c19">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c22">l2</font> = <font color="Maroon" title="c21">l1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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>  ;<br/>
<a NAME="E13:39_1_1_1_2_1"/><i><font color="Green" title="E21">A16</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">l</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c21">l1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green" title="E22">A17</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">xx</font>
<b>by </b><i><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E13:39_1_1_1_2_1"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">xx</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E13:39_1_1_1_2_1"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">xx</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E8:39_1_1_1_2_1"><i><font color="Green" title="E20">A15</font></i></a>, <a class="txt" href="abian.html#E13:39_1_1_1_2_1"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E13:39_1_1_1_2_1"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c21">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c21">l1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="txt" href="abian.html#E13:39_1_1_1_2_1"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c22">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
;<br/>
<a NAME="E16:39_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c15">yc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c2</font>
 
<b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_1_2_1"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="abian.html#E5:39_1_1_1_2_1"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="abian.html#E14:39_1_1_1_2_1"><i><font color="Green" title="E22">A17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:39_1_1_1_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
 <b>by </b><i><a class="txt" href="abian.html#E1:39_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="abian.html#E2:39_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:39_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_1_1_3"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c9">c2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">c1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">y</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="E1:39_1_1_1_3_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c9">c2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1_3_1"/><i><font color="Green" title="E18">A18</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c2</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">xx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font><b> such that </b><br/><a NAME="E5:39_1_1_1_3_1"/><i><font color="Green" title="E19">A19</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xx</font> &amp;  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being   <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>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; not <font color="Olive" title="b1">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Olive" title="b2">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) )
 <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_1_3_1"><i><font color="Green" title="E18">A18</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">yc</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_1_3_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="abian.html#E5:39_1_1_1_3_1"><i><font color="Green" title="E19">A19</font></i></a>, <a class="ref" href="abian.html#T6" target="_self" title="ABIAN:th.6">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">k</font>, <font color="Maroon" title="c17">l</font> being    <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> <b> such that </b><br/><a NAME="E8:39_1_1_1_3_1"/><i><font color="Green" title="E20">A20</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; not <font color="Maroon" title="c16">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c17">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 <b>by </b><i><a class="txt" href="abian.html#E5:39_1_1_1_3_1"><i><font color="Green" title="E19">A19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c16">k</font> as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  <b>by </b><i><a class="txt" href="abian.html#E8:39_1_1_1_3_1"><i><font color="Green" title="E20">A20</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">l</font> = <font color="Maroon" title="c17">l</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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>  <b>by </b><i><a class="txt" href="abian.html#E8:39_1_1_1_3_1"><i><font color="Green" title="E20">A20</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">k1</font> = <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c21">l1</font> = <font color="Maroon" title="c19">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c22">l2</font> = <font color="Maroon" title="c21">l1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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>  ;<br/>
<a NAME="E14:39_1_1_1_3_1"/><i><font color="Green" title="E21">A21</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">l</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c21">l1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green" title="E22">A22</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">xx</font>
<b>by </b><i><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E14:39_1_1_1_3_1"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">xx</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E14:39_1_1_1_3_1"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">xx</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E8:39_1_1_1_3_1"><i><font color="Green" title="E20">A20</font></i></a>, <a class="txt" href="abian.html#E14:39_1_1_1_3_1"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E14:39_1_1_1_3_1"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c21">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c21">l1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="txt" href="abian.html#E14:39_1_1_1_3_1"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c22">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
;<br/>
<a NAME="E17:39_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c15">yc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font>
 
<b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_1_3_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="abian.html#E5:39_1_1_1_3_1"><i><font color="Green" title="E19">A19</font></i></a>, <a class="txt" href="abian.html#E15:39_1_1_1_3_1"><i><font color="Green" title="E22">A22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:39_1_1_1_3_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1_1_3"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 <b>by </b><i><a class="txt" href="abian.html#E1:39_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="abian.html#E2:39_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E6:39_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
 <b>by </b><i><a class="txt" href="abian.html#E2:39_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_2"/>
<font color="Maroon" title="c7">c1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">c2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_2"/><i><font color="Green" title="E16">A23</font></i>: 
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">x11</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font><b> such that </b><br/><a NAME="E5:39_1_1_2"/><i><font color="Green" title="E17">A24</font></i>: 
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x11</font> &amp;  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being   <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>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Olive" title="b1">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Olive" title="b2">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) )
 <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2"><i><font color="Green" title="E16">A23</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c13">x12</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font><b> such that </b><br/><a NAME="E7:39_1_1_2"/><i><font color="Green" title="E18">A25</font></i>: 
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x12</font> &amp;  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being   <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>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; not <font color="Olive" title="b1">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Olive" title="b2">l</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) )
 <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2"><i><font color="Green" title="E16">A23</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c11">x1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="txt" href="abian.html#E5:39_1_1_2"><i><font color="Green" title="E17">A24</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">k1</font>, <font color="Maroon" title="c16">l1</font> being    <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> <b> such that </b><br/><a NAME="E10:39_1_1_2"/><i><font color="Green" title="E19">A26</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">k1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Maroon" title="c15">k1</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c16">l1</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 <b>by </b><i><a class="txt" href="abian.html#E5:39_1_1_2"><i><font color="Green" title="E17">A24</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">k2</font>, <font color="Maroon" title="c18">l2</font> being    <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> <b> such that </b><br/><a NAME="E12:39_1_1_2"/><i><font color="Green" title="E20">A27</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font> &amp; not <font color="Maroon" title="c17">k2</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c18">l2</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 <b>by </b><i><a class="txt" href="abian.html#E7:39_1_1_2"><i><font color="Green" title="E18">A25</font></i></a></i>;<br/><a NAME="E13:39_1_1_2"/><i><font color="Green" title="E21">A28</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">k1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">l1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><i><font color="Green" title="E22">A29</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">l1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">l1</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E7:39_1_1_2"><i><font color="Green" title="E18">A25</font></i></a>, <a class="txt" href="abian.html#E12:39_1_1_2"><i><font color="Green" title="E20">A27</font></i></a>, <a class="txt" href="abian.html#E13:39_1_1_2"><i><font color="Green" title="E21">A28</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">l1</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="txt" href="abian.html#E13:39_1_1_2"><i><font color="Green" title="E21">A28</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">l1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
;<br/><i><font color="Green" title="E23">A30</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">k1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l2</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">k1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E5:39_1_1_2"><i><font color="Green" title="E17">A24</font></i></a>, <a class="txt" href="abian.html#E10:39_1_1_2"><i><font color="Green" title="E19">A26</font></i></a>, <a class="txt" href="abian.html#E13:39_1_1_2"><i><font color="Green" title="E21">A28</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l2</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">l1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="txt" href="abian.html#E13:39_1_1_2"><i><font color="Green" title="E21">A28</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">l1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
;<br/><a NAME="E16:39_1_1_2"/>
 ex <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">k</font> being <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c19">k1</font> = <font color="Maroon" title="c15">k1</font>, <font color="Maroon" title="c20">l1</font> = <font color="Maroon" title="c16">l1</font>, <font color="Maroon" title="c21">l2</font> = <font color="Maroon" title="c18">l2</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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>  <b>by </b><i><a class="txt" href="abian.html#E10:39_1_1_2"><i><font color="Green" title="E19">A26</font></i></a>, <a class="txt" href="abian.html#E12:39_1_1_2"><i><font color="Green" title="E20">A27</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">k2</font> = <font color="Maroon" title="c17">k2</font> as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  <b>by </b><i><a class="txt" href="abian.html#E12:39_1_1_2"><i><font color="Green" title="E20">A27</font></i></a></i>;<br/>
<a NAME="E3:39_1_1_2_3"/><i><font color="Green" title="E24">A31</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_2_3_1"/>
( <font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font> or <font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_2_3_1_1"/><i><font color="Green" title="E25">A32</font></i>: 
<font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c23">r</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font>;<br/><b>reconsider </b><font color="Maroon" title="c24">k</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font></span>)</span> as    <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>  <b>by </b><i><a class="txt" href="abian.html#E1:39_1_1_2_3_1_1"><i><font color="Green" title="E25">A32</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">k</font> = <font color="Maroon" title="c24">k</font> as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  ;<br/><b>take </b>
<font color="Maroon" title="c25">k</font>
;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">k</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font>
<b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2_3"><i><font color="Green" title="E24">A31</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font></span>)</span>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font>
<b>by </b><i><a class="txt" href="abian.html#E14:39_1_1_2"><i><font color="Green" title="E22">A29</font></i></a>, <a class="txt" href="abian.html#E15:39_1_1_2"><i><font color="Green" title="E23">A30</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:39_1_1_2_3_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">r</font>
 ;<br/><b>thus </b><a NAME="E6:39_1_1_2_3_1_1"/>
<font color="Maroon" title="c23">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="ref" href="abian.html#T7" target="_self" title="ABIAN:th.7">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_2_3_1_2"/><i><font color="Green" title="E25">A33</font></i>: 
<font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c23">r</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font>;<br/><b>reconsider </b><font color="Maroon" title="c24">k</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">l2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font></span>)</span> as    <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>  <b>by </b><i><a class="txt" href="abian.html#E1:39_1_1_2_3_1_2"><i><font color="Green" title="E25">A33</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">k</font> = <font color="Maroon" title="c24">k</font> as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  ;<br/><b>take </b>
<font color="Maroon" title="c25">k</font>
;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">k</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font>
<b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2_3"><i><font color="Green" title="E24">A31</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font></span>)</span>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c22">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font>
<b>by </b><i><a class="txt" href="abian.html#E14:39_1_1_2"><i><font color="Green" title="E22">A29</font></i></a>, <a class="txt" href="abian.html#E15:39_1_1_2"><i><font color="Green" title="E23">A30</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:39_1_1_2_3_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">r</font>
 ;<br/><b>thus </b><a NAME="E6:39_1_1_2_3_1_2"/>
<font color="Maroon" title="c23">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="ref" href="abian.html#T7" target="_self" title="ABIAN:th.7">Th7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c19">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c20">k</font> being  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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> <b> such that </b><br/><a NAME="E18:39_1_1_2"/><i><font color="Green" title="E24">A34</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">r</font> &amp; <font color="Maroon" title="c19">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c21">r</font> = <font color="Maroon" title="c19">r</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="txt" href="abian.html#E18:39_1_1_2"><i><font color="Green" title="E24">A34</font></i></a></i>;<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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span>}</span> ;<br/><a NAME="E20:39_1_1_2"/><i><font color="Green" title="E25">A35</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> holds  <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <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> ] <b>means </b><font color="Maroon">H<sub>1</sub></font>($1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font>;<br/>
<a NAME="E1:39_1_1_2_4"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">l</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<b>hence </b><a NAME="E2:39_1_1_2_4"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c22">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c22">Odl</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">c</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E22:39_1_1_2"/><i><font color="Green" title="E26">A36</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> holds  <font color="Maroon" title="c22">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="abian.html#E20:39_1_1_2"><i><font color="Green" title="E25">A35</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_2_5"/>
<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">Odl</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c24">nc</font> = <font color="Maroon" title="c23">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:39_1_1_2_5"/><i><font color="Green" title="E27">A37</font></i>: 
<font color="Maroon" title="c22">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">nc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">nc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="abian.html#E22:39_1_1_2"><i><font color="Green" title="E26">A36</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c25">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:39_1_1_2_5"/><i><font color="Green" title="E28">A38</font></i>: 
( <font color="Maroon" title="c25">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c25">x'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><a NAME="E6:39_1_1_2_5"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">nc</font>,<font color="Maroon" title="c21">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="abian.html#E5:39_1_1_2_5"><i><font color="Green" title="E28">A38</font></i></a>, <a class="ref" href="eqrel_1.html#T30" title="EQREL_1:th.30">EQREL_1:30</a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">kn</font>, <font color="Maroon" title="c27">ln</font> being    <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> <b> such that </b><br/><a NAME="E8:39_1_1_2_5"/><i><font color="Green" title="E29">A39</font></i>: 
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c26">kn</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">nc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c27">ln</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="ref" href="abian.html#D7" target="_self" title="ABIAN:def.7">Def7</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <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> ] <b>means </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> $1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font>;<br/><a NAME="E9:39_1_1_2_5"/><i><font color="Green" title="E30">A40</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 <b>by </b><i><a class="ref" href="abian.html#T3" target="_self" title="ABIAN:th.3">Th3</a></i>;<br/><div><i><font color="Green" title="E31">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c28">i</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> ;<br/><b>assume </b><a NAME="E1:39_1_1_2_5_1"/><i><font color="Green" title="E32">A42</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c28">i</font>]
 ;<br/><a NAME="E2:39_1_1_2_5_1"/><i><font color="Green" title="E33">A43</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font> = 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c28">i</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">r</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon" title="c21">r</font>
<b>by </b><i><a class="txt" href="abian.html#E18:39_1_1_2"><i><font color="Green" title="E24">A34</font></i></a>, <a class="txt" href="abian.html#E1:39_1_1_2_5_1"><i><font color="Green" title="E32">A42</font></i></a>, <a class="txt" href="abian.html#E2:39_1_1_2_5_1"><i><font color="Green" title="E33">A43</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E4:39_1_1_2_5_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/></div><b>end;</b></div><a NAME="E11:39_1_1_2_5"/><i><font color="Green" title="E32">A44</font></i>: 
for <font color="Olive" title="b1">i</font> being   <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>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="abian.html#E9:39_1_1_2_5"><i><font color="Green" title="E30">A40</font></i></a>, <a class="txt" href="abian.html#E10:39_1_1_2_5"><i><font color="Green" title="E31">A41</font></i></a>);<br/></i><a NAME="E12:39_1_1_2_5"/>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">k</font>
 ;<br/><a NAME="E13:39_1_1_2_5"/><b>then </b><i><font color="Green" title="E33">A45</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><b>set </b><font color="Maroon" title="c28">dk</font> = <font color="Maroon" title="c27">ln</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c20">k</font>;<br/><b>set </b><font color="Maroon" title="c29">mk</font> = <font color="Maroon" title="c27">ln</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c20">k</font>;<br/><a NAME="E14:39_1_1_2_5"/><i><font color="Green" title="E34">A46</font></i>: 
<font color="Maroon" title="c27">ln</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">ln</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c20">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ln</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c20">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E13:39_1_1_2_5"><i><font color="Green" title="E33">A45</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/><a NAME="E15:39_1_1_2_5"/>
<font color="Maroon" title="c27">ln</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c20">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">k</font>
 <b>by </b><i><a class="txt" href="abian.html#E13:39_1_1_2_5"><i><font color="Green" title="E33">A45</font></i></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">kmk</font> = <font color="Maroon" title="c20">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ln</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c20">k</font></span>)</span> as    <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>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E17:39_1_1_2_5"/><i><font color="Green" title="E35">A47</font></i>: 
<font color="Maroon" title="c27">ln</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c30">kmk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">ln</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c20">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E14:39_1_1_2_5"><i><font color="Green" title="E34">A46</font></i></a></i>;<br/><a NAME="E18:39_1_1_2_5"/><i><font color="Green" title="E36">A48</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c26">kn</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c27">ln</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c30">kmk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c26">kn</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">nc</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c30">kmk</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c26">kn</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">nc</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c30">kmk</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c27">ln</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E8:39_1_1_2_5"><i><font color="Green" title="E29">A39</font></i></a>, <a class="txt" href="abian.html#E18:39_1_1_2_5"><i><font color="Green" title="E36">A48</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c30">kmk</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c27">ln</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">r</font>
<b>by </b><i><a class="txt" href="abian.html#E18:39_1_1_2_5"><i><font color="Green" title="E36">A48</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">ln</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c20">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font>
<b>by </b><i><a class="txt" href="abian.html#E17:39_1_1_2_5"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon" title="c21">r</font>
<b>by </b><i><a class="txt" href="abian.html#E11:39_1_1_2_5"><i><font color="Green" title="E32">A44</font></i></a></i>
;<br/><a NAME="E20:39_1_1_2_5"/><b>then </b>
<font color="Maroon" title="c26">kn</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c30">kmk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Odl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2_5"><i><font color="Green" title="E27">A37</font></i></a></i>;<br/><b>hence </b><a NAME="E21:39_1_1_2_5"/>
not <font color="Maroon" title="c22">Odl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c23">Odl</font> = <font color="Maroon" title="c22">Odl</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">c</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font>) <b>-&gt; </b>   <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 href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c24">odl</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">c</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E26:39_1_1_2"/><i><font color="Green" title="E27">A49</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> holds  <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon" title="c24">odl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>;<br/><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>not <font color="Maroon" title="c24">odl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>;<br/><b>set </b><font color="Maroon" title="c25">c1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c26">d2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c27">d1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E27:39_1_1_2"/><i><font color="Green" title="E28">A50</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">c</font>
 <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i><a NAME="E28:39_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="abian.html#E27:39_1_1_2"><i><font color="Green" title="E28">A50</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c28">d1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E30:39_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">c</font>
 <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i><b>then reconsider </b><font color="Maroon" title="c29">d2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">d3</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>take </b><font color="Maroon" title="c31">b</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c28">d1</font>,<font color="Maroon" title="c29">d2</font>,<font color="Maroon" title="c30">d3</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/><a NAME="E33:39_1_1_2"/><i><font color="Green" title="E29">A51</font></i>: 
( <font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">d1</font> &amp; <font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">d2</font> &amp; <font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">d3</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T47" title="MCART_1:th.47">MCART_1:47</a></i>;<br/><a NAME="E34:39_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(0) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="ref" href="abian.html#T3" target="_self" title="ABIAN:th.3">Th3</a></i>;<br/><a NAME="E35:39_1_1_2"/><b>then </b>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span>}</span> 
 ;<br/><a NAME="E36:39_1_1_2"/><b>then </b>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="txt" href="abian.html#E22:39_1_1_2"><i><font color="Green" title="E26">A36</font></i></a></i>;<br/><a NAME="E37:39_1_1_2"/><b>then </b>
 <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="abian.html#T2" target="_self" title="ABIAN:th.2">Th2</a></i>;<br/><a NAME="E38:39_1_1_2"/><b>then </b><i><font color="Green" title="E30">A52</font></i>: 
<font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0
 <b>by </b><i><a class="txt" href="abian.html#E26:39_1_1_2"><i><font color="Green" title="E27">A49</font></i></a></i>;<br/><a NAME="E39:39_1_1_2"/><b>then </b><i><font color="Green" title="E31">A53</font></i>: 
<font color="Maroon" title="c21">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 ;<br/><i><font color="Green" title="E32">A54</font></i>: <font color="Maroon" title="c28">d1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c30">d3</font> = 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c30">d3</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
<b>by </b><i><a class="txt" href="abian.html#E39:39_1_1_2"><i><font color="Green" title="E31">A53</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/><a NAME="E41:39_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c29">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_7"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c32">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_2_7_1"/><i><font color="Green" title="E33">A55</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c29">d2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_2_7_1_1"/>
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  or <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font> )
 </span><b>by </b><i><a class="txt" href="abian.html#E1:39_1_1_2_7_1"><i><font color="Green" title="E33">A55</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_7_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_2_7_1_1_1"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c33">xc</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font><b> such that </b><br/><a NAME="E3:39_1_1_2_7_1_1_1"/><i><font color="Green" title="E34">A56</font></i>: 
( <font color="Maroon" title="c33">xc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">x</font> &amp; <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">xc</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 ;<br/><b>thus </b><a NAME="E4:39_1_1_2_7_1_1_1"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2_7_1_1_1"><i><font color="Green" title="E34">A56</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_7_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c33">xc</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font><b> such that </b><br/><a NAME="E3:39_1_1_2_7_1_1_2"/><i><font color="Green" title="E34">A57</font></i>: 
( <font color="Maroon" title="c33">xc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">x</font> &amp; not <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">xc</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 ;<br/><b>thus </b><a NAME="E4:39_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2_7_1_1_2"><i><font color="Green" title="E34">A57</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c32">x</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="E2:39_1_1_2_7"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c33">xc</font> = <font color="Maroon" title="c32">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> ;<br/>
<a NAME="E4:39_1_1_2_7"/>
( <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">xc</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">xc</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 
;<br/>
<a NAME="E5:39_1_1_2_7"/><b>then </b>
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  or <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font> )
 
;<br/>
<b>hence </b><a NAME="E6:39_1_1_2_7"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c29">d2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E42:39_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="abian.html#E33:39_1_1_2"><i><font color="Green" title="E29">A51</font></i></a>, <a class="txt" href="abian.html#E40:39_1_1_2"><i><font color="Green" title="E32">A54</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/><a NAME="E43:39_1_1_2"/><i><font color="Green" title="E33">A58</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c29">d2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1_2_8"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c29">d2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c32">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_2_8"/><i><font color="Green" title="E34">A59</font></i>: 
( <font color="Maroon" title="c32">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  &amp; <font color="Maroon" title="c32">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:39_1_1_2_8"/>
(  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> st <br/>( <font color="Maroon" title="c32">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) &amp;  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> st <br/>( <font color="Maroon" title="c32">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; not <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) )
 
<b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2_8"><i><font color="Green" title="E34">A59</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:39_1_1_2_8"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E44:39_1_1_2"/><i><font color="Green" title="E34">A60</font></i>: 
<font color="Maroon" title="c28">d1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c29">d2</font>
 <b>by </b><i><a class="txt" href="abian.html#E43:39_1_1_2"><i><font color="Green" title="E33">A58</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/><a NAME="E45:39_1_1_2"/><i><font color="Green" title="E35">A61</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">r</font> holds <br/><font color="Maroon" title="c24">odl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c32">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font>;<br/>

<b>assume </b><a NAME="E1:39_1_1_2_9"/><i><font color="Green" title="E36">A62</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">r</font>
 ;<br/>

<div><i><font color="Green" title="E37">A63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_1_2_9_1"/>
<font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:39_1_1_2_9_1"/><b>then </b>
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E26:39_1_1_2"><i><font color="Green" title="E27">A49</font></i></a></i>;<br/><a NAME="E3:39_1_1_2_9_1"/><b>then </b>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#D8" title="CQC_SIM1:def.8">CQC_SIM1:def 8</a></i>;<br/><a NAME="E4:39_1_1_2_9_1"/><b>then </b>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="abian.html#E22:39_1_1_2"><i><font color="Green" title="E26">A36</font></i></a></i>;<br/><a NAME="E5:39_1_1_2_9_1"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <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>  st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> )
 ;<br/><b>hence </b><a NAME="E6:39_1_1_2_9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E1:39_1_1_2_9"><i><font color="Green" title="E36">A62</font></i></a>, <a class="ref" href="abian.html#T3" target="_self" title="ABIAN:th.3">Th3</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c33">fx</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="ref" href="abian.html#T6" target="_self" title="ABIAN:th.6">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">ofx</font> = <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">fx</font>, <font color="Maroon" title="c35">ox</font> = <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font> as    <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>  ;<br/>
<a NAME="E5:39_1_1_2_9"/>
<font color="Maroon" title="c35">ox</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="abian.html#E2:39_1_1_2_9"><i><font color="Green" title="E37">A63</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c36">ox1</font> = <font color="Maroon" title="c35">ox</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<a NAME="E7:39_1_1_2_9"/><i><font color="Green" title="E38">A64</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E39">A65</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c36">ox1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">fx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c36">ox1</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c36">ox1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c35">ox</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font>

;<br/>
<a NAME="E9:39_1_1_2_9"/>
<font color="Maroon" title="c35">ox</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E26:39_1_1_2"><i><font color="Green" title="E27">A49</font></i></a></i>;<br/>
<a NAME="E10:39_1_1_2_9"/><b>then </b>
<font color="Maroon" title="c35">ox</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8" title="CQC_SIM1:def.8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E11:39_1_1_2_9"/><b>then </b>
<font color="Maroon" title="c35">ox</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="abian.html#E22:39_1_1_2"><i><font color="Green" title="E26">A36</font></i></a></i>;<br/>
<a NAME="E12:39_1_1_2_9"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <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>  st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">ox</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> )
 
;<br/>
<a NAME="E13:39_1_1_2_9"/><b>then </b>
<font color="Maroon" title="c36">ox1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="abian.html#E8:39_1_1_2_9"><i><font color="Green" title="E39">A65</font></i></a></i>;<br/>
<a NAME="E14:39_1_1_2_9"/><b>then </b><i><font color="Green" title="E40">A66</font></i>: 
<font color="Maroon" title="c36">ox1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">fx</font>
 
<b>by </b><i><a class="txt" href="abian.html#E22:39_1_1_2"><i><font color="Green" title="E26">A36</font></i></a></i>;<br/>
<a NAME="E15:39_1_1_2_9"/>
<font color="Maroon" title="c34">ofx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">fx</font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E26:39_1_1_2"><i><font color="Green" title="E27">A49</font></i></a></i>;<br/>
<a NAME="E16:39_1_1_2_9"/><b>then </b><i><font color="Green" title="E41">A67</font></i>: 
<font color="Maroon" title="c34">ofx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">ox</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="abian.html#E14:39_1_1_2_9"><i><font color="Green" title="E40">A66</font></i></a>, <a class="ref" href="cqc_sim1.html#D8" title="CQC_SIM1:def.8">CQC_SIM1:def 8</a></i>;<br/>
<i><font color="Green" title="E42">A68</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c34">ofx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c34">ofx</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c34">ofx</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">fx</font>
<b>by </b><i><a class="txt" href="abian.html#E7:39_1_1_2_9"><i><font color="Green" title="E38">A64</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E18:39_1_1_2_9"/>
<font color="Maroon" title="c34">ofx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">fx</font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E26:39_1_1_2"><i><font color="Green" title="E27">A49</font></i></a></i>;<br/>
<a NAME="E19:39_1_1_2_9"/><b>then </b>
<font color="Maroon" title="c34">ofx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">fx</font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8" title="CQC_SIM1:def.8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E20:39_1_1_2_9"/><b>then </b>
<font color="Maroon" title="c34">ofx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="abian.html#E22:39_1_1_2"><i><font color="Green" title="E26">A36</font></i></a></i>;<br/>
<a NAME="E21:39_1_1_2_9"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <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>  st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">ofx</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> )
 
;<br/>
<a NAME="E22:39_1_1_2_9"/><b>then </b>
<font color="Maroon" title="c34">ofx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <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>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="abian.html#E17:39_1_1_2_9"><i><font color="Green" title="E42">A68</font></i></a></i>;<br/>
<a NAME="E23:39_1_1_2_9"/><b>then </b><i><font color="Green" title="E43">A69</font></i>: 
<font color="Maroon" title="c34">ofx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font>
 
<b>by </b><i><a class="txt" href="abian.html#E22:39_1_1_2"><i><font color="Green" title="E26">A36</font></i></a></i>;<br/>
<a NAME="E24:39_1_1_2_9"/>
<font color="Maroon" title="c35">ox</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E26:39_1_1_2"><i><font color="Green" title="E27">A49</font></i></a></i>;<br/>
<a NAME="E25:39_1_1_2_9"/><b>then </b>
<font color="Maroon" title="c34">ofx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">ox</font>
 
<b>by </b><i><a class="txt" href="abian.html#E23:39_1_1_2_9"><i><font color="Green" title="E43">A69</font></i></a>, <a class="ref" href="cqc_sim1.html#D8" title="CQC_SIM1:def.8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E26:39_1_1_2_9"/><b>then </b>
<font color="Maroon" title="c34">ofx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">ox</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E27:39_1_1_2_9"/>
<font color="Maroon" title="c24">odl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="abian.html#E16:39_1_1_2_9"><i><font color="Green" title="E41">A67</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E46:39_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_10"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_1_2_10"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c28">d1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">d2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c32">y</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="E1:39_1_1_2_10_1"/>
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c28">d1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c33">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_2_10_1"/><i><font color="Green" title="E36">A70</font></i>: 
( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">d1</font> &amp; <font color="Maroon" title="c32">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:39_1_1_2_10_1"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2_10_1"><i><font color="Green" title="E36">A70</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c34">xx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font><b> such that </b><br/><a NAME="E6:39_1_1_2_10_1"/><i><font color="Green" title="E37">A71</font></i>: 
( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">xx</font> &amp; <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">xx</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c35">ox</font> = <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">xx</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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>  <b>by </b><i><a class="txt" href="abian.html#E6:39_1_1_2_10_1"><i><font color="Green" title="E37">A71</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c36">yc</font> = <font color="Maroon" title="c32">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2_10_1"><i><font color="Green" title="E36">A70</font></i></a>, <a class="txt" href="abian.html#E6:39_1_1_2_10_1"><i><font color="Green" title="E37">A71</font></i></a>, <a class="ref" href="abian.html#T6" target="_self" title="ABIAN:th.6">Th6</a></i>;<br/>
<a NAME="E9:39_1_1_2_10_1"/>
<font color="Maroon" title="c21">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c34">xx</font>
 
<b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2_10_1"><i><font color="Green" title="E36">A70</font></i></a>, <a class="txt" href="abian.html#E6:39_1_1_2_10_1"><i><font color="Green" title="E37">A71</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E10:39_1_1_2_10_1"/><b>then </b>
<font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">yc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">ox</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="abian.html#E45:39_1_1_2"><i><font color="Green" title="E35">A61</font></i></a>, <a class="txt" href="abian.html#E3:39_1_1_2_10_1"><i><font color="Green" title="E36">A70</font></i></a>, <a class="txt" href="abian.html#E6:39_1_1_2_10_1"><i><font color="Green" title="E37">A71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:39_1_1_2_10_1"/>
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1_2_10"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
 <b>by </b><i><a class="txt" href="abian.html#E33:39_1_1_2"><i><font color="Green" title="E29">A51</font></i></a>, <a class="txt" href="abian.html#E44:39_1_1_2"><i><font color="Green" title="E34">A60</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E47:39_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_11"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_1_2_11"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c29">d2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c32">y</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="E1:39_1_1_2_11_1"/>
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c29">d2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c33">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_2_11_1"/><i><font color="Green" title="E36">A72</font></i>: 
( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font> &amp; <font color="Maroon" title="c32">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c34">xx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font><b> such that </b><br/><a NAME="E5:39_1_1_2_11_1"/><i><font color="Green" title="E37">A73</font></i>: 
( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">xx</font> &amp; not <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">xx</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2_11_1"><i><font color="Green" title="E36">A72</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c35">ox</font> = <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">xx</font> as  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <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>  <b>by </b><i><a class="txt" href="abian.html#E5:39_1_1_2_11_1"><i><font color="Green" title="E37">A73</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c36">yc</font> = <font color="Maroon" title="c32">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> <b>by </b><i><a class="txt" href="abian.html#E3:39_1_1_2_11_1"><i><font color="Green" title="E36">A72</font></i></a>, <a class="txt" href="abian.html#E5:39_1_1_2_11_1"><i><font color="Green" title="E37">A73</font></i></a>, <a class="ref" href="abian.html#T6" target="_self" title="ABIAN:th.6">Th6</a></i>;<br/>
<a NAME="E8:39_1_1_2_11_1"/>
<font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">yc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">ox</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="abian.html#E38:39_1_1_2"><i><font color="Green" title="E30">A52</font></i></a>, <a class="txt" href="abian.html#E45:39_1_1_2"><i><font color="Green" title="E35">A61</font></i></a>, <a class="txt" href="abian.html#E3:39_1_1_2_11_1"><i><font color="Green" title="E36">A72</font></i></a>, <a class="txt" href="abian.html#E5:39_1_1_2_11_1"><i><font color="Green" title="E37">A73</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:39_1_1_2_11_1"/>
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1_2_11"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 <b>by </b><i><a class="txt" href="abian.html#E33:39_1_1_2"><i><font color="Green" title="E29">A51</font></i></a>, <a class="txt" href="abian.html#E43:39_1_1_2"><i><font color="Green" title="E33">A58</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E48:39_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1_2_12"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c32">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_2_12"/><i><font color="Green" title="E36">A74</font></i>: 
( <font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> &amp; <font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:39_1_1_2_12"/><i><font color="Green" title="E37">A75</font></i>: 
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font>
 
<b>by </b><i><a class="txt" href="abian.html#E33:39_1_1_2"><i><font color="Green" title="E29">A51</font></i></a>, <a class="txt" href="abian.html#E3:39_1_1_2_12"><i><font color="Green" title="E36">A74</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c33">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:39_1_1_2_12"/><i><font color="Green" title="E38">A76</font></i>: 
( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c32">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> )
 <b>by </b><i><a class="txt" href="abian.html#E33:39_1_1_2"><i><font color="Green" title="E29">A51</font></i></a>, <a class="txt" href="abian.html#E3:39_1_1_2_12"><i><font color="Green" title="E36">A74</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:39_1_1_2_12"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font>
 
<b>by </b><i><a class="txt" href="abian.html#E6:39_1_1_2_12"><i><font color="Green" title="E38">A76</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:39_1_1_2_12"/><b>then </b>
<font color="Maroon" title="c21">r</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:39_1_1_2_12"><i><font color="Green" title="E37">A75</font></i></a>, <a class="txt" href="abian.html#E6:39_1_1_2_12"><i><font color="Green" title="E38">A76</font></i></a>, <a class="ref" href="abian.html#D4" target="_self" title="ABIAN:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E9:39_1_1_2_12"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E1:39"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="abian.html#D5" target="_self" title="ABIAN:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span><b> such that </b><br/><a NAME="E4:39"/><i><font color="Green" title="E15">A77</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="abian.html#E2:39"><i><font color="Green" title="E14">A2</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c4">E1c</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c5">E1</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c6">E2c</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c7">E2</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c8">E3c</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c9">E3</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>take </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
;<br/>

<b>take </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
;<br/>

<b>take </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
;<br/>

<b>thus </b><a NAME="E5:39"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 ;<br/><a NAME="E2:39_2_1"/><b>then </b><i><font color="Green" title="E16">A78</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_2_1_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="abian.html#E2:39_2_1"><i><font color="Green" title="E16">A78</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1"><b>suppose </b></a><a NAME="E1:39_2_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_2_1_1_1"/><i><font color="Green" title="E17">A79</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span><b> such that </b><br/><a NAME="E5:39_2_1_1_1"/><i><font color="Green" title="E18">A80</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
 <b>by </b><i><a class="txt" href="abian.html#E3:39_2_1_1_1"><i><font color="Green" title="E17">A79</font></i></a></i>;<br/><b>thus </b><a NAME="E6:39_2_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="txt" href="abian.html#E3:39_2_1_1_1"><i><font color="Green" title="E17">A79</font></i></a>, <a class="txt" href="abian.html#E5:39_2_1_1_1"><i><font color="Green" title="E18">A80</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_2"><b>suppose </b></a><a NAME="E1:39_2_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_2_1_1_2"/><i><font color="Green" title="E17">A81</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span><b> such that </b><br/><a NAME="E5:39_2_1_1_2"/><i><font color="Green" title="E18">A82</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 <b>by </b><i><a class="txt" href="abian.html#E3:39_2_1_1_2"><i><font color="Green" title="E17">A81</font></i></a></i>;<br/><b>thus </b><a NAME="E6:39_2_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="txt" href="abian.html#E3:39_2_1_1_2"><i><font color="Green" title="E17">A81</font></i></a>, <a class="txt" href="abian.html#E5:39_2_1_1_2"><i><font color="Green" title="E18">A82</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_3"><b>suppose </b></a><a NAME="E1:39_2_1_1_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_2_1_1_3"/><i><font color="Green" title="E17">A83</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span><b> such that </b><br/><a NAME="E5:39_2_1_1_3"/><i><font color="Green" title="E18">A84</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
 <b>by </b><i><a class="txt" href="abian.html#E3:39_2_1_1_3"><i><font color="Green" title="E17">A83</font></i></a></i>;<br/><b>thus </b><a NAME="E6:39_2_1_1_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="txt" href="abian.html#E3:39_2_1_1_3"><i><font color="Green" title="E17">A83</font></i></a>, <a class="txt" href="abian.html#E5:39_2_1_1_3"><i><font color="Green" title="E18">A84</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c10">x</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="E2:39_2"/><i><font color="Green" title="E16">A85</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c11">c</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">x</font>;<br/>
<a NAME="E3:39_2"/><i><font color="Green" title="E17">A86</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="abian.html#E2:39_2"><i><font color="Green" title="E16">A85</font></i></a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">c</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> <b>by </b><i><a class="txt" href="abian.html#E2:39_2"><i><font color="Green" title="E16">A85</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E5:39_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E4:39"><i><font color="Green" title="E15">A77</font></i></a>, <a class="txt" href="abian.html#E3:39_2"><i><font color="Green" title="E17">A86</font></i></a></i>;<br/>
<a NAME="E6:39_2"/><b>then </b><i><font color="Green" title="E18">A87</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_2_2"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  )
 </span><b>by </b><i><a class="txt" href="abian.html#E6:39_2"><i><font color="Green" title="E18">A87</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_2_1"><b>suppose </b></a><a NAME="E1:39_2_2_1"/><i><font color="Green" title="E19">A88</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
 ;<br/><div class="add"><a NAME="E2:39_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><a NAME="E3:39_2_2_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="abian.html#E1:39_2_2_1"><i><font color="Green" title="E19">A88</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E4:39_2_2_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:39_2_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_2_2"><b>suppose </b></a><a NAME="E1:39_2_2_2"/><i><font color="Green" title="E19">A89</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 ;<br/><div class="add"><a NAME="E2:39_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><a NAME="E3:39_2_2_2"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="abian.html#E1:39_2_2_2"><i><font color="Green" title="E19">A89</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E4:39_2_2_2"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:39_2_2_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_2_3"><b>suppose </b></a><a NAME="E1:39_2_2_3"/><i><font color="Green" title="E19">A90</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
 ;<br/><div class="add"><a NAME="E2:39_2_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><a NAME="E3:39_2_2_3"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="abian.html#E1:39_2_2_3"><i><font color="Green" title="E19">A90</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E4:39_2_2_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E6:39"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_3"/>
not <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_3"/><i><font color="Green" title="E16">A91</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:39_3"/><i><font color="Green" title="E17">A92</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E3:39_3"><i><font color="Green" title="E16">A91</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span><b> such that </b><br/><a NAME="E7:39_3"/><i><font color="Green" title="E18">A93</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
 <b>by </b><i><a class="txt" href="abian.html#E5:39_3"><i><font color="Green" title="E17">A92</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:39_3"/><i><font color="Green" title="E19">A94</font></i>: 
( <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xx</font> )
 <b>by </b><i><a class="txt" href="abian.html#E3:39_3"><i><font color="Green" title="E16">A91</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">YY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:39_3"/><i><font color="Green" title="E20">A95</font></i>: 
( <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">YY</font> &amp; <font color="Maroon" title="c14">YY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E9:39_3"><i><font color="Green" title="E19">A94</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">cc</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span><b> such that </b><br/><a NAME="E13:39_3"/><i><font color="Green" title="E21">A96</font></i>: 
<font color="Maroon" title="c14">YY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
 <b>by </b><i><a class="txt" href="abian.html#E11:39_3"><i><font color="Green" title="E20">A95</font></i></a></i>;<br/>
<a NAME="E14:39_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_3"><i><font color="Green" title="E17">A92</font></i></a>, <a class="txt" href="abian.html#E7:39_3"><i><font color="Green" title="E18">A93</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:39_3"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:39_3"/><b>then </b><i><font color="Green" title="E22">A97</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">c</font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:39"><i><font color="Green" title="E15">A77</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:39_3"/><i><font color="Green" title="E23">A98</font></i>: 
( <font color="Maroon" title="c16">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c16">x'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E19:39_3"/><i><font color="Green" title="E24">A99</font></i>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="abian.html#E16:39_3"><i><font color="Green" title="E22">A97</font></i></a>, <a class="txt" href="abian.html#E18:39_3"><i><font color="Green" title="E23">A98</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E20:39_3"/>
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E11:39_3"><i><font color="Green" title="E20">A95</font></i></a>, <a class="txt" href="abian.html#E13:39_3"><i><font color="Green" title="E21">A96</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E21:39_3"/><b>then </b>
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E22:39_3"/><b>then </b><i><font color="Green" title="E25">A100</font></i>: 
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">cc</font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:39"><i><font color="Green" title="E15">A77</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">xx'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E24:39_3"/><i><font color="Green" title="E26">A101</font></i>: 
( <font color="Maroon" title="c17">xx'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c15">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c17">xx'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E25:39_3"/><i><font color="Green" title="E27">A102</font></i>: 
<font color="Maroon" title="c15">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c13">xx</font>
 
<b>by </b><i><a class="txt" href="abian.html#E22:39_3"><i><font color="Green" title="E25">A100</font></i></a>, <a class="txt" href="abian.html#E24:39_3"><i><font color="Green" title="E26">A101</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E26:39_3"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:39_3"/><b>then </b><i><font color="Green" title="E28">A103</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_3"><i><font color="Green" title="E17">A92</font></i></a>, <a class="txt" href="abian.html#E7:39_3"><i><font color="Green" title="E18">A93</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xx</font> = 
<font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="abian.html#E9:39_3"><i><font color="Green" title="E19">A94</font></i></a>, <a class="ref" href="funct_7.html#T72" title="FUNCT_7:th.72">FUNCT_7:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="abian.html#E27:39_3"><i><font color="Green" title="E28">A103</font></i></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,0</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T70" title="FUNCT_7:th.70">FUNCT_7:70</a></i>
;<br/>
<a NAME="E29:39_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c13">xx</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_3"><i><font color="Green" title="E17">A92</font></i></a>, <a class="txt" href="abian.html#E7:39_3"><i><font color="Green" title="E18">A93</font></i></a>, <a class="txt" href="abian.html#E11:39_3"><i><font color="Green" title="E20">A95</font></i></a>, <a class="txt" href="abian.html#E13:39_3"><i><font color="Green" title="E21">A96</font></i></a>, <a class="ref" href="abian.html#D7" target="_self" title="ABIAN:def.7">Def7</a></i>;<br/>
<a NAME="E30:39_3"/><b>then </b><i><font color="Green" title="E29">A104</font></i>: 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c13">xx</font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_3"><i><font color="Green" title="E17">A92</font></i></a>, <a class="txt" href="abian.html#E7:39_3"><i><font color="Green" title="E18">A93</font></i></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>;<br/>
<a NAME="E31:39_3"/><i><font color="Green" title="E30">A105</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c14">YY</font>
 
<b>by </b><i><a class="txt" href="abian.html#E9:39_3"><i><font color="Green" title="E19">A94</font></i></a>, <a class="txt" href="abian.html#E11:39_3"><i><font color="Green" title="E20">A95</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E32:39_3"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c14">YY</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c14">YY</font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:39"><i><font color="Green" title="E15">A77</font></i></a>, <a class="txt" href="abian.html#E13:39_3"><i><font color="Green" title="E21">A96</font></i></a></i>;<br/>
<b>hence </b><a NAME="E33:39_3"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E5:39_3"><i><font color="Green" title="E17">A92</font></i></a>, <a class="txt" href="abian.html#E7:39_3"><i><font color="Green" title="E18">A93</font></i></a>, <a class="txt" href="abian.html#E9:39_3"><i><font color="Green" title="E19">A94</font></i></a>, <a class="txt" href="abian.html#E13:39_3"><i><font color="Green" title="E21">A96</font></i></a>, <a class="txt" href="abian.html#E19:39_3"><i><font color="Green" title="E24">A99</font></i></a>, <a class="txt" href="abian.html#E25:39_3"><i><font color="Green" title="E27">A102</font></i></a>, <a class="txt" href="abian.html#E30:39_3"><i><font color="Green" title="E29">A104</font></i></a>, <a class="txt" href="abian.html#E31:39_3"><i><font color="Green" title="E30">A105</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:39"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_4"/>
not <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_4"/><i><font color="Green" title="E16">A106</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:39_4"/><i><font color="Green" title="E17">A107</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E3:39_4"><i><font color="Green" title="E16">A106</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span><b> such that </b><br/><a NAME="E7:39_4"/><i><font color="Green" title="E18">A108</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 <b>by </b><i><a class="txt" href="abian.html#E5:39_4"><i><font color="Green" title="E17">A107</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:39_4"/><i><font color="Green" title="E19">A109</font></i>: 
( <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xx</font> )
 <b>by </b><i><a class="txt" href="abian.html#E3:39_4"><i><font color="Green" title="E16">A106</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">YY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:39_4"/><i><font color="Green" title="E20">A110</font></i>: 
( <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">YY</font> &amp; <font color="Maroon" title="c14">YY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E9:39_4"><i><font color="Green" title="E19">A109</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">cc</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span><b> such that </b><br/><a NAME="E13:39_4"/><i><font color="Green" title="E21">A111</font></i>: 
<font color="Maroon" title="c14">YY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 <b>by </b><i><a class="txt" href="abian.html#E11:39_4"><i><font color="Green" title="E20">A110</font></i></a></i>;<br/>
<a NAME="E14:39_4"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_4"><i><font color="Green" title="E17">A107</font></i></a>, <a class="txt" href="abian.html#E7:39_4"><i><font color="Green" title="E18">A108</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:39_4"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:39_4"/><b>then </b><i><font color="Green" title="E22">A112</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">c</font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:39"><i><font color="Green" title="E15">A77</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:39_4"/><i><font color="Green" title="E23">A113</font></i>: 
( <font color="Maroon" title="c16">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c16">x'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E19:39_4"/><i><font color="Green" title="E24">A114</font></i>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="abian.html#E16:39_4"><i><font color="Green" title="E22">A112</font></i></a>, <a class="txt" href="abian.html#E18:39_4"><i><font color="Green" title="E23">A113</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E20:39_4"/>
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E11:39_4"><i><font color="Green" title="E20">A110</font></i></a>, <a class="txt" href="abian.html#E13:39_4"><i><font color="Green" title="E21">A111</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E21:39_4"/><b>then </b>
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E22:39_4"/><b>then </b><i><font color="Green" title="E25">A115</font></i>: 
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">cc</font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:39"><i><font color="Green" title="E15">A77</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">xx'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E24:39_4"/><i><font color="Green" title="E26">A116</font></i>: 
( <font color="Maroon" title="c17">xx'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c15">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c17">xx'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E25:39_4"/><i><font color="Green" title="E27">A117</font></i>: 
<font color="Maroon" title="c15">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c13">xx</font>
 
<b>by </b><i><a class="txt" href="abian.html#E22:39_4"><i><font color="Green" title="E25">A115</font></i></a>, <a class="txt" href="abian.html#E24:39_4"><i><font color="Green" title="E26">A116</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E26:39_4"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:39_4"/><b>then </b><i><font color="Green" title="E28">A118</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_4"><i><font color="Green" title="E17">A107</font></i></a>, <a class="txt" href="abian.html#E7:39_4"><i><font color="Green" title="E18">A108</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xx</font> = 
<font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="abian.html#E9:39_4"><i><font color="Green" title="E19">A109</font></i></a>, <a class="ref" href="funct_7.html#T72" title="FUNCT_7:th.72">FUNCT_7:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="abian.html#E27:39_4"><i><font color="Green" title="E28">A118</font></i></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,0</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T70" title="FUNCT_7:th.70">FUNCT_7:70</a></i>
;<br/>
<a NAME="E29:39_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c13">xx</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_4"><i><font color="Green" title="E17">A107</font></i></a>, <a class="txt" href="abian.html#E7:39_4"><i><font color="Green" title="E18">A108</font></i></a>, <a class="txt" href="abian.html#E11:39_4"><i><font color="Green" title="E20">A110</font></i></a>, <a class="txt" href="abian.html#E13:39_4"><i><font color="Green" title="E21">A111</font></i></a>, <a class="ref" href="abian.html#D7" target="_self" title="ABIAN:def.7">Def7</a></i>;<br/>
<a NAME="E30:39_4"/><b>then </b><i><font color="Green" title="E29">A119</font></i>: 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c13">xx</font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_4"><i><font color="Green" title="E17">A107</font></i></a>, <a class="txt" href="abian.html#E7:39_4"><i><font color="Green" title="E18">A108</font></i></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>;<br/>
<a NAME="E31:39_4"/><i><font color="Green" title="E30">A120</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c14">YY</font>
 
<b>by </b><i><a class="txt" href="abian.html#E9:39_4"><i><font color="Green" title="E19">A109</font></i></a>, <a class="txt" href="abian.html#E11:39_4"><i><font color="Green" title="E20">A110</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E32:39_4"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c14">YY</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c14">YY</font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:39"><i><font color="Green" title="E15">A77</font></i></a>, <a class="txt" href="abian.html#E13:39_4"><i><font color="Green" title="E21">A111</font></i></a></i>;<br/>
<b>hence </b><a NAME="E33:39_4"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E5:39_4"><i><font color="Green" title="E17">A107</font></i></a>, <a class="txt" href="abian.html#E7:39_4"><i><font color="Green" title="E18">A108</font></i></a>, <a class="txt" href="abian.html#E9:39_4"><i><font color="Green" title="E19">A109</font></i></a>, <a class="txt" href="abian.html#E13:39_4"><i><font color="Green" title="E21">A111</font></i></a>, <a class="txt" href="abian.html#E19:39_4"><i><font color="Green" title="E24">A114</font></i></a>, <a class="txt" href="abian.html#E25:39_4"><i><font color="Green" title="E27">A117</font></i></a>, <a class="txt" href="abian.html#E30:39_4"><i><font color="Green" title="E29">A119</font></i></a>, <a class="txt" href="abian.html#E31:39_4"><i><font color="Green" title="E30">A120</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E8:39"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_5"/>
not <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_5"/><i><font color="Green" title="E16">A121</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span> </span>)</span> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:39_5"/><i><font color="Green" title="E17">A122</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E3:39_5"><i><font color="Green" title="E16">A121</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span><b> such that </b><br/><a NAME="E7:39_5"/><i><font color="Green" title="E18">A123</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
 <b>by </b><i><a class="txt" href="abian.html#E5:39_5"><i><font color="Green" title="E17">A122</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:39_5"/><i><font color="Green" title="E19">A124</font></i>: 
( <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xx</font> )
 <b>by </b><i><a class="txt" href="abian.html#E3:39_5"><i><font color="Green" title="E16">A121</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">YY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:39_5"/><i><font color="Green" title="E20">A125</font></i>: 
( <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">YY</font> &amp; <font color="Maroon" title="c14">YY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E9:39_5"><i><font color="Green" title="E19">A124</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">cc</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span><b> such that </b><br/><a NAME="E13:39_5"/><i><font color="Green" title="E21">A126</font></i>: 
<font color="Maroon" title="c14">YY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
 <b>by </b><i><a class="txt" href="abian.html#E11:39_5"><i><font color="Green" title="E20">A125</font></i></a></i>;<br/>
<a NAME="E14:39_5"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_5"><i><font color="Green" title="E17">A122</font></i></a>, <a class="txt" href="abian.html#E7:39_5"><i><font color="Green" title="E18">A123</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:39_5"/><b>then </b><i><font color="Green" title="E22">A127</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">c</font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:39"><i><font color="Green" title="E15">A77</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:39_5"/><i><font color="Green" title="E23">A128</font></i>: 
( <font color="Maroon" title="c16">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c16">x'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E18:39_5"/><i><font color="Green" title="E24">A129</font></i>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="abian.html#E15:39_5"><i><font color="Green" title="E22">A127</font></i></a>, <a class="txt" href="abian.html#E17:39_5"><i><font color="Green" title="E23">A128</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E19:39_5"/>
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E11:39_5"><i><font color="Green" title="E20">A125</font></i></a>, <a class="txt" href="abian.html#E13:39_5"><i><font color="Green" title="E21">A126</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E20:39_5"/><b>then </b><i><font color="Green" title="E25">A130</font></i>: 
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">cc</font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:39"><i><font color="Green" title="E15">A77</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">xx'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E22:39_5"/><i><font color="Green" title="E26">A131</font></i>: 
( <font color="Maroon" title="c17">xx'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c15">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c17">xx'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E23:39_5"/><i><font color="Green" title="E27">A132</font></i>: 
<font color="Maroon" title="c15">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c13">xx</font>
 
<b>by </b><i><a class="txt" href="abian.html#E20:39_5"><i><font color="Green" title="E25">A130</font></i></a>, <a class="txt" href="abian.html#E22:39_5"><i><font color="Green" title="E26">A131</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E24:39_5"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:39_5"/><b>then </b><i><font color="Green" title="E28">A133</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_5"><i><font color="Green" title="E17">A122</font></i></a>, <a class="txt" href="abian.html#E7:39_5"><i><font color="Green" title="E18">A123</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xx</font> = 
<font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="abian.html#E9:39_5"><i><font color="Green" title="E19">A124</font></i></a>, <a class="ref" href="funct_7.html#T72" title="FUNCT_7:th.72">FUNCT_7:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="abian.html#E25:39_5"><i><font color="Green" title="E28">A133</font></i></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,0</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T70" title="FUNCT_7:th.70">FUNCT_7:70</a></i>
;<br/>
<a NAME="E27:39_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c13">xx</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_5"><i><font color="Green" title="E17">A122</font></i></a>, <a class="txt" href="abian.html#E7:39_5"><i><font color="Green" title="E18">A123</font></i></a>, <a class="txt" href="abian.html#E11:39_5"><i><font color="Green" title="E20">A125</font></i></a>, <a class="txt" href="abian.html#E13:39_5"><i><font color="Green" title="E21">A126</font></i></a>, <a class="ref" href="abian.html#D7" target="_self" title="ABIAN:def.7">Def7</a></i>;<br/>
<a NAME="E28:39_5"/><b>then </b><i><font color="Green" title="E29">A134</font></i>: 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c13">xx</font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:39_5"><i><font color="Green" title="E17">A122</font></i></a>, <a class="txt" href="abian.html#E7:39_5"><i><font color="Green" title="E18">A123</font></i></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>;<br/>
<a NAME="E29:39_5"/><i><font color="Green" title="E30">A135</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c14">YY</font>
 
<b>by </b><i><a class="txt" href="abian.html#E9:39_5"><i><font color="Green" title="E19">A124</font></i></a>, <a class="txt" href="abian.html#E11:39_5"><i><font color="Green" title="E20">A125</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E30:39_5"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c14">YY</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c14">YY</font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:39"><i><font color="Green" title="E15">A77</font></i></a>, <a class="txt" href="abian.html#E13:39_5"><i><font color="Green" title="E21">A126</font></i></a></i>;<br/>
<b>hence </b><a NAME="E31:39_5"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E5:39_5"><i><font color="Green" title="E17">A122</font></i></a>, <a class="txt" href="abian.html#E7:39_5"><i><font color="Green" title="E18">A123</font></i></a>, <a class="txt" href="abian.html#E9:39_5"><i><font color="Green" title="E19">A124</font></i></a>, <a class="txt" href="abian.html#E13:39_5"><i><font color="Green" title="E21">A126</font></i></a>, <a class="txt" href="abian.html#E18:39_5"><i><font color="Green" title="E24">A129</font></i></a>, <a class="txt" href="abian.html#E23:39_5"><i><font color="Green" title="E27">A132</font></i></a>, <a class="txt" href="abian.html#E28:39_5"><i><font color="Green" title="E29">A134</font></i></a>, <a class="txt" href="abian.html#E29:39_5"><i><font color="Green" title="E30">A135</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>


</div>
