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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <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>  st <font color="Olive" title="b1">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a>  holds <br/> ex <font color="Olive" title="b2">E1</font>, <font color="Olive" title="b3">E2</font>, <font color="Olive" title="b4">E3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">E1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b3">E2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b4">E3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Olive" title="b1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b2">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">E1</font> &amp; <font color="Olive" title="b1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b3">E2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">E2</font> &amp; <font color="Olive" title="b1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b4">E3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">E3</font> )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a>  implies  ex <font color="Olive" title="b1">E1</font>, <font color="Olive" title="b2">E2</font>, <font color="Olive" title="b3">E3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">E1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">E2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b3">E3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">E1</font> &amp; <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b2">E2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">E2</font> &amp; <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b3">E3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">E3</font> ) )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <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="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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>] <span class="kw">means </span>( <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/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <span class="kw">-&gt; </span>  <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/>
<span class="kw">assume </span><a NAME="E1:39"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">E1</font>, <font color="Olive" title="b2">E2</font>, <font color="Olive" title="b3">E3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">E1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">E2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b3">E3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">E1</font> &amp; <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b2">E2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">E2</font> &amp; <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b3">E3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">E3</font> )</span><br/>

<a NAME="E2:39"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
 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#K8" title="EQREL_1:func.8">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#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</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"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c3">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">c</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">b</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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="c5">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">x0</font> = <font color="Maroon" title="c5">x0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T20" title="EQREL_1:th.20">EQREL_1:20</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c6">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/>
<span class="kw">set </span><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="c4">c</font> : <font color="Maroon">S<sub>2</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="c4">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="c4">c</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">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="c4">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c6">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/>
<span class="kw">set </span><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="c4">c</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> ;<br/>
<a NAME="E8:39_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="c4">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="c4">c</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">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="c4">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:39_1_1"/>
( <font color="Maroon" title="c8">c1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">c2</font> or <font color="Maroon" title="c8">c1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">c2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c8">c1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">b</font>]</span><br/><div class="add"><span class="kw">take </span><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="c8">c1</font>,<font color="Maroon" title="c10">c2</font>,<font color="Maroon" title="c3">c3</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">b</font>]</span><br/><a NAME="E2:39_1_1_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<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="c10">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>;<br/><a NAME="E3:39_1_1_1"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<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="c3">c3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>;<br/><a NAME="E4:39_1_1_1"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<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="c8">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>;<br/><span class="kw">thus </span><a NAME="E5:39_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="c4">c</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  &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"><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>  &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"><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>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><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> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> )</span><br/><span class="kw">assume </span><a NAME="E1:39_1_1_1_1_1"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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="c8">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="c10">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="c3">c3</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E2:39_1_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E3:39_1_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E1:39_1_1_1_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_1"><span class="kw">suppose </span></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="c8">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font></span><br/><div class="add"><a NAME="E2:39_1_1_1_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">xx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font> st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">xx</font> &amp;  ex <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font> &amp; <font color="Olive" title="b2">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Olive" title="b3">l</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) )
 ;<br/><span class="kw">hence </span><a NAME="E3:39_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="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_2"><span class="kw">suppose </span></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="c10">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font></span><br/><div class="add"><a NAME="E2:39_1_1_1_1_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">xx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font> st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">xx</font> &amp;  ex <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font> &amp;  not <font color="Olive" title="b2">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Olive" title="b3">l</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) )
 ;<br/><span class="kw">hence </span><a NAME="E3:39_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="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1_3"><span class="kw">suppose </span></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="c3">c3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font></span><br/><div class="add"><span class="kw">hence </span><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="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> or <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> )</span><br/>

<span class="kw">assume </span><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="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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="c4">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="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T19" title="EQREL_1:th.19">EQREL_1:19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">k</font>, <font color="Maroon" title="c15">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:39_1_1_1_1"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<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="c6">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D7" target="_self" title="ABIAN:def.7">Def7</a></span>;<br/>
<a NAME="E7:39_1_1_1_1"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E8:39_1_1_1_1"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<font color="Maroon" title="c14">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">thus </span><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>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><a NAME="E2:39_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E1:39_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E25">A13</font></span></a></span>;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E3:39_1_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1_1_1_2"><span class="kw">suppose </span></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">l</font> = <font color="Maroon" title="c15">l</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">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="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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
<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="c6">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E8:39_1_1_1_1"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<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="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_1_1_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">l</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="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
;<br/><a NAME="E4:39_1_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c2</font>
 ;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E3:39_1_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
 not <font color="Maroon" title="c14">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">k</font> = <font color="Maroon" title="c14">k</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">thus </span><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>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><a NAME="E2:39_1_1_1_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E1:39_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E25">A14</font></span></a></span>;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E3:39_1_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_2_1_1_2"><span class="kw">suppose </span></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">l</font> = <font color="Maroon" title="c15">l</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">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="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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
<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="c6">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E8:39_1_1_1_1"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<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="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_1_1_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">l</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="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
;<br/><a NAME="E4:39_1_1_1_1_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">c1</font>
 ;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E3:39_1_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><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> <font color="Maroon" title="c8">c1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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="c8">c1</font> or <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c2</font> )</span><br/>

<a NAME="E1:39_1_1_1_2"/><span class="lab"><font color="Green" title="E22">A15</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:39_1_1_1_2"/>
<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="c8">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:39_1_1_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E5:39_1_1_1_2"/><span class="lab"><font color="Green" title="E23">A16</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">c1</font>
 <span class="kw">and </span><br/><a NAME="E6:39_1_1_1_2"/><span class="lab"><font color="Green" title="E24">A17</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">consider </span><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="c4">c</font><span class="kw"> such that </span><br/><a NAME="E8:39_1_1_1_2"/><span class="lab"><font color="Green" title="E25">A18</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xx</font>
 <span class="kw">and </span><br/><a NAME="E9:39_1_1_1_2"/><span class="lab"><font color="Green" title="E26">A19</font></span>: 
 ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c6">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_1_1_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">k</font>, <font color="Maroon" title="c16">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E11:39_1_1_1_2"/><span class="lab"><font color="Green" title="E27">A20</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">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="c16">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font>
 <span class="kw">and </span><br/><a NAME="E12:39_1_1_1_2"/><span class="lab"><font color="Green" title="E28">A21</font></span>: 
( <font color="Maroon" title="c15">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c16">l</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_1_1_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">k</font> = <font color="Maroon" title="c15">k</font>, <font color="Maroon" title="c18">l</font> = <font color="Maroon" title="c16">l</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39_1_1_1_2"><span class="lab"><font color="Green" title="E28">A21</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">k1</font> = <font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">l1</font> = <font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">l2</font> = <font color="Maroon" title="c20">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E17:39_1_1_1_2"/><span class="lab"><font color="Green" title="E29">A22</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">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="c4">c</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_1_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E8:39_1_1_1_2"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="ref" href="abian.html#T6" target="_self" title="ABIAN:th.6">Th6</a></span>;<br/>
<a NAME="E19:39_1_1_1_2"/><span class="lab"><font color="Green" title="E30">A23</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E20:39_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A24</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E32">A25</font></span>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k1</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="c14">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T69" title="FUNCT_7:th.69">FUNCT_7:69</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
;<br/>
<a NAME="E22:39_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A26</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">l1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k1</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="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="c19">k1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">xx</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
<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="c19">k1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_1_1_1_2"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<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="c17">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
<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="c18">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39_1_1_1_2"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="txt" href="#E17:39_1_1_1_2"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<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">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1_1_1_2"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<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">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
<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="c20">l1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:39_1_1_1_2"><span class="lab"><font color="Green" title="E33">A26</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c21">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
;<br/>
<a NAME="E24:39_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">yc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_1_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E8:39_1_1_1_2"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="txt" href="#E21:39_1_1_1_2"><span class="lab"><font color="Green" title="E32">A25</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:39_1_1_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:39_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E4:39_1_1_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  &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"><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>  )</span><br/><a NAME="E8:39_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c10">c2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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="c10">c2</font> or <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">c1</font> )</span><br/>

<a NAME="E1:39_1_1_1_3"/><span class="lab"><font color="Green" title="E22">A27</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:39_1_1_1_3"/>
<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="c10">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">c1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:39_1_1_1_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E5:39_1_1_1_3"/><span class="lab"><font color="Green" title="E23">A28</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c2</font>
 <span class="kw">and </span><br/><a NAME="E6:39_1_1_1_3"/><span class="lab"><font color="Green" title="E24">A29</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">consider </span><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="c4">c</font><span class="kw"> such that </span><br/><a NAME="E8:39_1_1_1_3"/><span class="lab"><font color="Green" title="E25">A30</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xx</font>
 <span class="kw">and </span><br/><a NAME="E9:39_1_1_1_3"/><span class="lab"><font color="Green" title="E26">A31</font></span>: 
 ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c6">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_1_1_3"><span class="lab"><font color="Green" title="E23">A28</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">k</font>, <font color="Maroon" title="c16">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E11:39_1_1_1_3"/><span class="lab"><font color="Green" title="E27">A32</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">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="c16">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font>
 <span class="kw">and </span><br/><a NAME="E12:39_1_1_1_3"/><span class="lab"><font color="Green" title="E28">A33</font></span>: 
 not <font color="Maroon" title="c15">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">and </span><br/><a NAME="E13:39_1_1_1_3"/><span class="lab"><font color="Green" title="E29">A34</font></span>: 
<font color="Maroon" title="c16">l</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_1_1_3"><span class="lab"><font color="Green" title="E26">A31</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">l</font> = <font color="Maroon" title="c16">l</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_1_1_1_3"><span class="lab"><font color="Green" title="E29">A34</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c15">k</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39_1_1_1_3"><span class="lab"><font color="Green" title="E28">A33</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">k1</font> = <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">l1</font> = <font color="Maroon" title="c17">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">l2</font> = <font color="Maroon" title="c20">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E19:39_1_1_1_3"/><span class="lab"><font color="Green" title="E30">A35</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">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="c4">c</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_1_3"><span class="lab"><font color="Green" title="E24">A29</font></span></a>, <a class="txt" href="#E8:39_1_1_1_3"><span class="lab"><font color="Green" title="E25">A30</font></span></a>, <a class="ref" href="abian.html#T6" target="_self" title="ABIAN:th.6">Th6</a></span>;<br/>
<a NAME="E21:39_1_1_1_3"/><span class="lab"><font color="Green" title="E31">A36</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">l</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E22:39_1_1_1_3"/><span class="lab"><font color="Green" title="E32">A37</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E33">A38</font></span>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k1</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="c14">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T69" title="FUNCT_7:th.69">FUNCT_7:69</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1_3"><span class="lab"><font color="Green" title="E22">A27</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
;<br/>
<a NAME="E24:39_1_1_1_3"/><span class="lab"><font color="Green" title="E34">A39</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">l1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k1</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="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="c19">k1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">xx</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
<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="c19">k1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:39_1_1_1_3"><span class="lab"><font color="Green" title="E32">A37</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
<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="c17">l</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39_1_1_1_3"><span class="lab"><font color="Green" title="E27">A32</font></span></a>, <a class="txt" href="#E19:39_1_1_1_3"><span class="lab"><font color="Green" title="E30">A35</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<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="c17">l</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:39_1_1_1_3"><span class="lab"><font color="Green" title="E31">A36</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<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">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
<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="c20">l1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:39_1_1_1_3"><span class="lab"><font color="Green" title="E34">A39</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c21">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
;<br/>
<a NAME="E26:39_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c22">yc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_1_3"><span class="lab"><font color="Green" title="E24">A29</font></span></a>, <a class="txt" href="#E8:39_1_1_1_3"><span class="lab"><font color="Green" title="E25">A30</font></span></a>, <a class="txt" href="#E23:39_1_1_1_3"><span class="lab"><font color="Green" title="E33">A38</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:39_1_1_1_3"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:39_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E2:39_1_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/><span class="kw">thus </span><a NAME="E10:39_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_2"/>
<font color="Maroon" title="c8">c1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">b</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1_1_2"/><span class="lab"><font color="Green" title="E18">A40</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">c1</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1_1_2"/><span class="lab"><font color="Green" title="E19">A41</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">consider </span><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="c4">c</font><span class="kw"> such that </span><br/><a NAME="E6:39_1_1_2"/><span class="lab"><font color="Green" title="E20">A42</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x11</font>
 <span class="kw">and </span><br/><a NAME="E7:39_1_1_2"/><span class="lab"><font color="Green" title="E21">A43</font></span>: 
 ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c6">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1_2"><span class="lab"><font color="Green" title="E18">A40</font></span></a></span>;<br/><span class="kw">consider </span><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="c4">c</font><span class="kw"> such that </span><br/><a NAME="E9:39_1_1_2"/><span class="lab"><font color="Green" title="E22">A44</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x12</font>
 <span class="kw">and </span><br/><a NAME="E10:39_1_1_2"/><span class="lab"><font color="Green" title="E23">A45</font></span>: 
 ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c6">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_2"><span class="lab"><font color="Green" title="E19">A41</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">k2</font>, <font color="Maroon" title="c15">l2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E12:39_1_1_2"/><span class="lab"><font color="Green" title="E24">A46</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c14">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="c15">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font>
 <span class="kw">and </span><br/><a NAME="E13:39_1_1_2"/><span class="lab"><font color="Green" title="E25">A47</font></span>: 
 not <font color="Maroon" title="c14">k2</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">and </span><br/><a NAME="E14:39_1_1_2"/><span class="lab"><font color="Green" title="E26">A48</font></span>: 
<font color="Maroon" title="c15">l2</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39_1_1_2"><span class="lab"><font color="Green" title="E23">A45</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">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="c4">c</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_2"><span class="lab"><font color="Green" title="E20">A42</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">k1</font>, <font color="Maroon" title="c18">l1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E17:39_1_1_2"/><span class="lab"><font color="Green" title="E27">A49</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">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="c18">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font>
 <span class="kw">and </span><br/><a NAME="E18:39_1_1_2"/><span class="lab"><font color="Green" title="E28">A50</font></span>: 
( <font color="Maroon" title="c17">k1</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c18">l1</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_1_2"><span class="lab"><font color="Green" title="E21">A43</font></span></a></span>;<br/><a NAME="E19:39_1_1_2"/><span class="lab"><font color="Green" title="E29">A51</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E20:39_1_1_2"/><span class="lab"><font color="Green" title="E30">A52</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E31">A53</font></span>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">l2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">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="c17">k1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T77" title="FUNCT_7:th.77">FUNCT_7:77</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">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="c18">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_2"><span class="lab"><font color="Green" title="E20">A42</font></span></a>, <a class="txt" href="#E17:39_1_1_2"><span class="lab"><font color="Green" title="E27">A49</font></span></a>, <a class="txt" href="#E19:39_1_1_2"><span class="lab"><font color="Green" title="E29">A51</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">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="c18">l1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_1_1_2"><span class="lab"><font color="Green" title="E30">A52</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T77" title="FUNCT_7:th.77">FUNCT_7:77</a></span>
;<br/><a NAME="E22:39_1_1_2"/><span class="lab"><font color="Green" title="E32">A54</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">l2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E23:39_1_1_2"/><span class="lab"><font color="Green" title="E33">A55</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c14">k2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E34">A56</font></span>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">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="c14">k2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T77" title="FUNCT_7:th.77">FUNCT_7:77</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">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="c15">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_1_2"><span class="lab"><font color="Green" title="E22">A44</font></span></a>, <a class="txt" href="#E12:39_1_1_2"><span class="lab"><font color="Green" title="E24">A46</font></span></a>, <a class="txt" href="#E23:39_1_1_2"><span class="lab"><font color="Green" title="E33">A55</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">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="c15">l2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:39_1_1_2"><span class="lab"><font color="Green" title="E32">A54</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T77" title="FUNCT_7:th.77">FUNCT_7:77</a></span>
;<br/><a NAME="E25:39_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#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c4">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c19">k2</font> = <font color="Maroon" title="c14">k2</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_1_1_2"><span class="lab"><font color="Green" title="E25">A47</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">k1</font> = <font color="Maroon" title="c17">k1</font>, <font color="Maroon" title="c21">l1</font> = <font color="Maroon" title="c18">l1</font>, <font color="Maroon" title="c22">l2</font> = <font color="Maroon" title="c15">l2</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1_1_2"><span class="lab"><font color="Green" title="E28">A50</font></span></a>, <a class="txt" href="#E14:39_1_1_2"><span class="lab"><font color="Green" title="E26">A48</font></span></a></span>;<br/>
<a NAME="E3:39_1_1_2_3"/><span class="lab"><font color="Green" title="E35">A57</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:39_1_1_2_3"/><span class="lab"><font color="Green" title="E36">A58</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">l1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:39_1_1_2_3_1"/>
( <font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">l1</font> or <font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c19">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">l1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_2_3_1_1"/>
<font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">l1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c4">c</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">k</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c19">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">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="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l2</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c24">r</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> &amp; <font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c25">k</font> = <font color="Maroon" title="c23">k</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">take </span>
<font color="Maroon" title="c25">k</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> &amp; <font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> )</span><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="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">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="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1_2_3"><span class="lab"><font color="Green" title="E35">A57</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l2</font></span>)</span>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T77" title="FUNCT_7:th.77">FUNCT_7:77</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:39_1_1_2"><span class="lab"><font color="Green" title="E34">A56</font></span></a>, <a class="txt" href="#E21:39_1_1_2"><span class="lab"><font color="Green" title="E31">A53</font></span></a></span>
;<br/><span class="kw">hence </span><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="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font></span><br/><span class="kw">thus </span><a NAME="E6:39_1_1_2_3_1_1"/>
<font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T7" target="_self" title="ABIAN:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_2_3_1_2"/>
<font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c19">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">l1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c4">c</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">k</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">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="c19">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">l1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c24">r</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> &amp; <font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c25">k</font> = <font color="Maroon" title="c23">k</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">take </span>
<font color="Maroon" title="c25">k</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> &amp; <font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> )</span><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">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">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">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <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="c16">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_2_3"><span class="lab"><font color="Green" title="E36">A58</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">l1</font></span>)</span>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T77" title="FUNCT_7:th.77">FUNCT_7:77</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">l1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:39_1_1_2"><span class="lab"><font color="Green" title="E34">A56</font></span></a>, <a class="txt" href="#E21:39_1_1_2"><span class="lab"><font color="Green" title="E31">A53</font></span></a></span>
;<br/><span class="kw">hence </span><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="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font></span><br/><span class="kw">thus </span><a NAME="E6:39_1_1_2_3_1_2"/>
<font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T7" target="_self" title="ABIAN:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><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#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E27:39_1_1_2"/><span class="lab"><font color="Green" title="E35">A59</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E28:39_1_1_2"/><span class="lab"><font color="Green" title="E36">A60</font></span>: 
<font color="Maroon" title="c19">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 ;<br/><span class="kw">reconsider </span><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="c4">c</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:39_1_1_2"><span class="lab"><font color="Green" title="E36">A60</font></span></a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <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#M1" title="SUBSET_1:mode.1">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="E30:39_1_1_2"/><span class="lab"><font color="Green" title="E37">A61</font></span>: 
 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="c4">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="setfam_1.html#K9" title="SETFAM_1:func.9">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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="c4">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><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#M1" title="SUBSET_1:mode.1">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>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span>
<span class="kw">hence </span><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="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><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="c4">c</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E32:39_1_1_2"/><span class="lab"><font color="Green" title="E38">A62</font></span>: 
 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="c4">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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="#E30:39_1_1_2"><span class="lab"><font color="Green" title="E37">A61</font></span></a>);<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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/><span class="kw">let </span><font color="Maroon" title="c23">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">Odl</font> implies  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>  )</span><br/><span class="kw">assume </span><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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">Odl</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><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="c4">c</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E3:39_1_1_2_5"/><span class="lab"><font color="Green" title="E39">A63</font></span>: 
<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#M1" title="SUBSET_1:mode.1">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:39_1_1_2"><span class="lab"><font color="Green" title="E38">A62</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E40">A64</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c25">i</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c25">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:39_1_1_2_5_1"/><span class="lab"><font color="Green" title="E41">A65</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c25">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c25">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><a NAME="E2:39_1_1_2_5_1"/><span class="lab"><font color="Green" title="E42">A66</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">i</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> = 
<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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c25">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T77" title="FUNCT_7:th.77">FUNCT_7:77</a></span>
<br/>.= 
<font color="Maroon" title="c21">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:39_1_1_2"><span class="lab"><font color="Green" title="E35">A59</font></span></a>, <a class="txt" href="#E1:39_1_1_2_5_1"><span class="lab"><font color="Green" title="E41">A65</font></span></a>, <a class="txt" href="#E2:39_1_1_2_5_1"><span class="lab"><font color="Green" title="E42">A66</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:39_1_1_2_5_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c25">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:39_1_1_2_5"/><span class="lab"><font color="Green" title="E41">A67</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T3" target="_self" title="ABIAN:th.3">Th3</a></span>;<br/><a NAME="E6:39_1_1_2_5"/><span class="lab"><font color="Green" title="E42">A68</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E5:39_1_1_2_5"><span class="lab"><font color="Green" title="E41">A67</font></span></a>, <a class="txt" href="#E4:39_1_1_2_5"><span class="lab"><font color="Green" title="E40">A64</font></span></a>);<br/></span><a NAME="E7:39_1_1_2_5"/>
 ex <font color="Olive" title="b1">x9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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="Olive" title="b1">x9</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/><a NAME="E8:39_1_1_2_5"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T22" title="EQREL_1:th.22">EQREL_1:22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">kn</font>, <font color="Maroon" title="c26">ln</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E10:39_1_1_2_5"/><span class="lab"><font color="Green" title="E43">A69</font></span>: 
<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="c25">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="c26">ln</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D7" target="_self" title="ABIAN:def.7">Def7</a></span>;<br/><a NAME="E11:39_1_1_2_5"/><span class="lab"><font color="Green" title="E44">A70</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c26">ln</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c27">mk</font> = <font color="Maroon" title="c26">ln</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c20">k</font>;<br/><span class="kw">set </span><font color="Maroon" title="c28">dk</font> = <font color="Maroon" title="c26">ln</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c20">k</font>;<br/><a NAME="E12:39_1_1_2_5"/><span class="lab"><font color="Green" title="E45">A71</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">kn</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E13:39_1_1_2_5"/><span class="lab"><font color="Green" title="E46">A72</font></span>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">k</font>
 ;<br/><a NAME="E14:39_1_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c26">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">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="c26">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><a NAME="E16:39_1_1_2_5"/>
<font color="Maroon" title="c26">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#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">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#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ln</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c20">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_1_1_2_5"><span class="lab"><font color="Green" title="E46">A72</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>;<br/><a NAME="E17:39_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A73</font></span>: 
<font color="Maroon" title="c26">ln</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">kmk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c29">kmk</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">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="c29">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="c25">kn</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">nc</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T77" title="FUNCT_7:th.77">FUNCT_7:77</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c29">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="c26">ln</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39_1_1_2_5"><span class="lab"><font color="Green" title="E43">A69</font></span></a>, <a class="txt" href="#E12:39_1_1_2_5"><span class="lab"><font color="Green" title="E45">A71</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c29">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">ln</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39_1_1_2_5"><span class="lab"><font color="Green" title="E44">A70</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">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#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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:39_1_1_2_5"><span class="lab"><font color="Green" title="E47">A73</font></span></a>, <a class="ref" href="funct_7.html#T77" title="FUNCT_7:th.77">FUNCT_7:77</a></span>
<br/>.= 
<font color="Maroon" title="c21">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_2_5"><span class="lab"><font color="Green" title="E42">A68</font></span></a></span>
;<br/><a NAME="E19:39_1_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c25">kn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1_2_5"><span class="lab"><font color="Green" title="E39">A63</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E20:39_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><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="c4">c</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D9" title="FUNCT_1:def.9">FUNCT_1:def 9</a></span>;<br/><span class="kw">deffunc </span><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="c4">c</font>) <span class="kw">-&gt; </span>   <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="seq_4.html#K6" title="SEQ_4:func.6">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/><span class="kw">consider </span><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="c4">c</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a><span class="kw"> such that </span><br/><a NAME="E36:39_1_1_2"/><span class="lab"><font color="Green" title="E39">A74</font></span>: 
 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="c4">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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font>] <span class="kw">means </span><font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ;<br/><span class="kw">set </span><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="c4">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> ;<br/><span class="kw">set </span><font color="Maroon" title="c26">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="c4">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="E37:39_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="c4">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="c4">c</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span><a NAME="E38:39_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A75</font></span>: 
<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="c4">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="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E39:39_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <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 class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T3" target="_self" title="ABIAN:th.3">Th3</a></span>;<br/><a NAME="E40:39_1_1_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#M1" title="SUBSET_1:mode.1">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="E41:39_1_1_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:39_1_1_2"><span class="lab"><font color="Green" title="E38">A62</font></span></a></span>;<br/><a NAME="E42:39_1_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K6" title="SEQ_4:func.6">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T2" target="_self" title="ABIAN:th.2">Th2</a></span>;<br/><a NAME="E43:39_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A76</font></span>: 
<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#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:39_1_1_2"><span class="lab"><font color="Green" title="E39">A74</font></span></a></span>;<br/><a NAME="E44:39_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A77</font></span>: 
<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="c4">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> 
 ;<br/><span class="kw">reconsider </span><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="c4">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:39_1_1_2"><span class="lab"><font color="Green" title="E40">A75</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font>] <span class="kw">means </span> not <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c28">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="c4">c</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> ;<br/><a NAME="E46:39_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="c4">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="c4">c</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span><span class="kw">then </span><span class="kw">reconsider </span><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="c4">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E48:39_1_1_2"/><span class="lab"><font color="Green" title="E43">A78</font></span>: 
 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="c4">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_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c30">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">r</font> implies <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="c30">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="c30">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c31">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="c30">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T6" target="_self" title="ABIAN:th.6">Th6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">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="c31">fx</font>, <font color="Maroon" title="c33">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="c30">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">assume </span><a NAME="E3:39_1_1_2_6"/><span class="lab"><font color="Green" title="E44">A79</font></span>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c30">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="c30">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_1_2_6_1"/>
<font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:39_1_1_2_6_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">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="c30">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:39_1_1_2"><span class="lab"><font color="Green" title="E39">A74</font></span></a></span>;<br/><a NAME="E3:39_1_1_2_6_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c30">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/><a NAME="E4:39_1_1_2_6_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#M1" title="SUBSET_1:mode.1">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="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:39_1_1_2"><span class="lab"><font color="Green" title="E38">A62</font></span></a></span>;<br/><a NAME="E5:39_1_1_2_6_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> )
 ;<br/><span class="kw">hence </span><a NAME="E6:39_1_1_2_6_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1_2_6"><span class="lab"><font color="Green" title="E44">A79</font></span></a>, <a class="ref" href="abian.html#T3" target="_self" title="ABIAN:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c34">ox1</font> = <font color="Maroon" title="c33">ox</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<a NAME="E6:39_1_1_2_6"/>
<font color="Maroon" title="c33">ox</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">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="c30">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:39_1_1_2"><span class="lab"><font color="Green" title="E39">A74</font></span></a></span>;<br/>
<a NAME="E7:39_1_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c33">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="c30">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/>
<a NAME="E8:39_1_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c33">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#M1" title="SUBSET_1:mode.1">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="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:39_1_1_2"><span class="lab"><font color="Green" title="E38">A62</font></span></a></span>;<br/>
<a NAME="E9:39_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A80</font></span>: 
 ex <font color="Olive" title="b1">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">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="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> )
 
;<br/>
<a NAME="E10:39_1_1_2_6"/><span class="lab"><font color="Green" title="E46">A81</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c34">ox1</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">fx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c34">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="c30">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c34">ox1</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="c30">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T69" title="FUNCT_7:th.69">FUNCT_7:69</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c33">ox</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">x</font>

;<br/>
<a NAME="E12:39_1_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c34">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#M1" title="SUBSET_1:mode.1">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="c31">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_1_2_6"><span class="lab"><font color="Green" title="E45">A80</font></span></a></span>;<br/>
<a NAME="E13:39_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A82</font></span>: 
<font color="Maroon" title="c34">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="c31">fx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:39_1_1_2"><span class="lab"><font color="Green" title="E38">A62</font></span></a></span>;<br/>
<a NAME="E14:39_1_1_2_6"/>
<font color="Maroon" title="c32">ofx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">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="c31">fx</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:39_1_1_2"><span class="lab"><font color="Green" title="E39">A74</font></span></a></span>;<br/>
<a NAME="E15:39_1_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c32">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="c31">fx</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/>
<a NAME="E16:39_1_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c32">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#M1" title="SUBSET_1:mode.1">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="c31">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:39_1_1_2"><span class="lab"><font color="Green" title="E38">A62</font></span></a></span>;<br/>
<a NAME="E17:39_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A83</font></span>: 
 ex <font color="Olive" title="b1">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">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="c31">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> )
 
;<br/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c32">ofx</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="c30">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c32">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="c30">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T69" title="FUNCT_7:th.69">FUNCT_7:69</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c32">ofx</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">fx</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39_1_1_2_6"><span class="lab"><font color="Green" title="E46">A81</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
;<br/>
<a NAME="E19:39_1_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c32">ofx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#M1" title="SUBSET_1:mode.1">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="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:39_1_1_2_6"><span class="lab"><font color="Green" title="E48">A83</font></span></a></span>;<br/>
<a NAME="E20:39_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A84</font></span>: 
<font color="Maroon" title="c32">ofx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c30">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:39_1_1_2"><span class="lab"><font color="Green" title="E38">A62</font></span></a></span>;<br/>
<a NAME="E21:39_1_1_2_6"/>
<font color="Maroon" title="c33">ox</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">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="c30">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:39_1_1_2"><span class="lab"><font color="Green" title="E39">A74</font></span></a></span>;<br/>
<a NAME="E22:39_1_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c32">ofx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c33">ox</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_1_1_2_6"><span class="lab"><font color="Green" title="E49">A84</font></span></a>, <a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/>
<a NAME="E23:39_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A85</font></span>: 
<font color="Maroon" title="c32">ofx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c33">ox</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/>
<a NAME="E24:39_1_1_2_6"/>
<font color="Maroon" title="c32">ofx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">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="c31">fx</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:39_1_1_2"><span class="lab"><font color="Green" title="E39">A74</font></span></a></span>;<br/>
<a NAME="E25:39_1_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c32">ofx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">ox</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_1_1_2_6"><span class="lab"><font color="Green" title="E47">A82</font></span></a>, <a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:39_1_1_2_6"/>
<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="c30">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="c30">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:39_1_1_2_6"><span class="lab"><font color="Green" title="E50">A85</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E49:39_1_1_2"/><span class="lab"><font color="Green" title="E44">A86</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c27">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_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c30">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c30">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="c27">d1</font> or <font color="Maroon" title="c30">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_1_2_7"/>
<font color="Maroon" title="c30">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="c27">d1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1_1_2_7"/>
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1_1_2_7"/><span class="lab"><font color="Green" title="E45">A87</font></span>: 
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">d1</font>
 <span class="kw">and </span><br/><a NAME="E5:39_1_1_2_7"/><span class="lab"><font color="Green" title="E46">A88</font></span>: 
<font color="Maroon" title="c30">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="c31">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E6:39_1_1_2_7"/>
<font color="Maroon" title="c31">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="c4">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_2_7"><span class="lab"><font color="Green" title="E45">A87</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">xx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font><span class="kw"> such that </span><br/><a NAME="E8:39_1_1_2_7"/><span class="lab"><font color="Green" title="E47">A89</font></span>: 
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">xx</font>
 <span class="kw">and </span><br/><a NAME="E9:39_1_1_2_7"/><span class="lab"><font color="Green" title="E48">A90</font></span>: 
<font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">xx</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c33">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">xx</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_1_2_7"><span class="lab"><font color="Green" title="E48">A90</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c34">yc</font> = <font color="Maroon" title="c30">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_1_2_7"><span class="lab"><font color="Green" title="E46">A88</font></span></a>, <a class="txt" href="#E8:39_1_1_2_7"><span class="lab"><font color="Green" title="E47">A89</font></span></a>, <a class="ref" href="abian.html#T6" target="_self" title="ABIAN:th.6">Th6</a></span>;<br/>
<a NAME="E12:39_1_1_2_7"/>
<font color="Maroon" title="c21">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c32">xx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_2_7"><span class="lab"><font color="Green" title="E45">A87</font></span></a>, <a class="txt" href="#E8:39_1_1_2_7"><span class="lab"><font color="Green" title="E47">A89</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/>
<a NAME="E13:39_1_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">yc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">ox</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:39_1_1_2"><span class="lab"><font color="Green" title="E43">A78</font></span></a>, <a class="txt" href="#E5:39_1_1_2_7"><span class="lab"><font color="Green" title="E46">A88</font></span></a>, <a class="txt" href="#E8:39_1_1_2_7"><span class="lab"><font color="Green" title="E47">A89</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:39_1_1_2_7"/>
<font color="Maroon" title="c30">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E50:39_1_1_2"/><span class="lab"><font color="Green" title="E45">A91</font></span>: 
<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="c4">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="c4">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_8"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><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> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">c</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="c4">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></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">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="c4">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> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> )</span><br/><span class="kw">assume </span><a NAME="E1:39_1_1_2_8_1"/><span class="lab"><font color="Green" title="E46">A92</font></span>: 
<font color="Maroon" title="c30">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="c4">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_8_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:39_1_1_2_8_1_1"/>
( <font color="Maroon" title="c30">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="c4">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span>  or <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_2_8_1"><span class="lab"><font color="Green" title="E46">A92</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_8_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_2_8_1_1_1"/>
<font color="Maroon" title="c30">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="c4">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font></span><br/><div class="add"><a NAME="E2:39_1_1_2_8_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">xc</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font> st <br/>( <font color="Olive" title="b1">xc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">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">xc</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E3:39_1_1_2_8_1_1_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_8_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_2_8_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font></span><br/><div class="add"><a NAME="E2:39_1_1_2_8_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">xc</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font> st <br/>( <font color="Olive" title="b1">xc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">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">xc</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E3:39_1_1_2_8_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> or <font color="Maroon" title="c30">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="c4">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> )</span><br/>

<span class="kw">assume </span><a NAME="E2:39_1_1_2_8"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">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="c4">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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">xc</font> = <font color="Maroon" title="c30">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">c</font> ;<br/>
<a NAME="E4:39_1_1_2_8"/>
( <font color="Maroon" title="c24">odl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">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="c31">xc</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 
;<br/>
<a NAME="E5:39_1_1_2_8"/><span class="kw">then </span>
( <font color="Maroon" title="c30">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="c4">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span>  or <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">d2</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E6:39_1_1_2_8"/>
<font color="Maroon" title="c30">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="c4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><span class="kw">take </span><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="c27">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c31">b</font>]</span><br/><a NAME="E52:39_1_1_2"/><span class="lab"><font color="Green" title="E46">A93</font></span>: 
<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="c27">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>;<br/><a NAME="E53:39_1_1_2"/><span class="lab"><font color="Green" title="E47">A94</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>;<br/><a NAME="E54:39_1_1_2"/><span class="lab"><font color="Green" title="E48">A95</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>;<br/><font color="Maroon" title="c27">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="c4">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>
<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="c4">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:39_1_1_2"><span class="lab"><font color="Green" title="E42">A77</font></span></a>, <a class="ref" href="zfmisc_1.html#T40" title="ZFMISC_1:th.40">ZFMISC_1:40</a></span>
;<br/><span class="kw">hence </span><a NAME="E56:39_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="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:39_1_1_2"><span class="lab"><font color="Green" title="E46">A93</font></span></a>, <a class="txt" href="#E53:39_1_1_2"><span class="lab"><font color="Green" title="E47">A94</font></span></a>, <a class="txt" href="#E54:39_1_1_2"><span class="lab"><font color="Green" title="E48">A95</font></span></a>, <a class="txt" href="#E50:39_1_1_2"><span class="lab"><font color="Green" title="E45">A91</font></span></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  &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"><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>  &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"><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>  )</span><br/><a NAME="E57:39_1_1_2"/><span class="lab"><font color="Green" title="E49">A96</font></span>: 
<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="c4">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_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_1_2_10"/>
<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="c4">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1_1_2_10"/><span class="lab"><font color="Green" title="E50">A97</font></span>: 
( <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="c4">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E4:39_1_1_2_10"/>
(  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="c4">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="c4">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>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1_2_10"><span class="lab"><font color="Green" title="E50">A97</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:39_1_1_2_10"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E58:39_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c27">d1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c29">d2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/><span class="kw">hence </span><a NAME="E59:39_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:39_1_1_2"><span class="lab"><font color="Green" title="E46">A93</font></span></a>, <a class="txt" href="#E49:39_1_1_2"><span class="lab"><font color="Green" title="E44">A86</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  &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"><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>  )</span><br/><a NAME="E60:39_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="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="c4">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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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="c4">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_1_2_11"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c4">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c33">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1_1_2_11"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1_1_2_11"/><span class="lab"><font color="Green" title="E50">A98</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:39_1_1_2_11"/><span class="lab"><font color="Green" title="E51">A99</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">consider </span><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="c4">c</font><span class="kw"> such that </span><br/><a NAME="E7:39_1_1_2_11"/><span class="lab"><font color="Green" title="E52">A100</font></span>: 
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">xx</font>
 <span class="kw">and </span><br/><a NAME="E8:39_1_1_2_11"/><span class="lab"><font color="Green" title="E53">A101</font></span>: 
 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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_2_11"><span class="lab"><font color="Green" title="E50">A98</font></span></a></span>;<br/>
<span class="kw">reconsider </span><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#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_1_1_2_11"><span class="lab"><font color="Green" title="E53">A101</font></span></a></span>;<br/>
<span class="kw">reconsider </span><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="c4">c</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_1_2_11"><span class="lab"><font color="Green" title="E51">A99</font></span></a>, <a class="txt" href="#E7:39_1_1_2_11"><span class="lab"><font color="Green" title="E52">A100</font></span></a>, <a class="ref" href="abian.html#T6" target="_self" title="ABIAN:th.6">Th6</a></span>;<br/>
<a NAME="E11:39_1_1_2_11"/>
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:39_1_1_2"><span class="lab"><font color="Green" title="E41">A76</font></span></a>, <a class="txt" href="#E48:39_1_1_2"><span class="lab"><font color="Green" title="E43">A78</font></span></a>, <a class="txt" href="#E5:39_1_1_2_11"><span class="lab"><font color="Green" title="E51">A99</font></span></a>, <a class="txt" href="#E7:39_1_1_2_11"><span class="lab"><font color="Green" title="E52">A100</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:39_1_1_2_11"/>
<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="c4">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E61:39_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:39_1_1_2"><span class="lab"><font color="Green" title="E47">A94</font></span></a>, <a class="txt" href="#E57:39_1_1_2"><span class="lab"><font color="Green" title="E49">A96</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/><span class="kw">thus </span><a NAME="E62:39_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> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1_1_2_12"/><span class="lab"><font color="Green" title="E50">A102</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:39_1_1_2_12"/><span class="lab"><font color="Green" title="E51">A103</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E5:39_1_1_2_12"/><span class="lab"><font color="Green" title="E52">A104</font></span>: 
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:39_1_1_2"><span class="lab"><font color="Green" title="E48">A95</font></span></a>, <a class="txt" href="#E4:39_1_1_2_12"><span class="lab"><font color="Green" title="E51">A103</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c33">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:39_1_1_2_12"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E8:39_1_1_2_12"/><span class="lab"><font color="Green" title="E53">A105</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E9:39_1_1_2_12"/><span class="lab"><font color="Green" title="E54">A106</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:39_1_1_2"><span class="lab"><font color="Green" title="E48">A95</font></span></a>, <a class="txt" href="#E3:39_1_1_2_12"><span class="lab"><font color="Green" title="E50">A102</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E10:39_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_1_1_2_12"><span class="lab"><font color="Green" title="E53">A105</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E11:39_1_1_2_12"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_1_2_12"><span class="lab"><font color="Green" title="E52">A104</font></span></a>, <a class="txt" href="#E9:39_1_1_2_12"><span class="lab"><font color="Green" title="E54">A106</font></span></a>, <a class="ref" href="abian.html#D4" target="_self" title="ABIAN:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:39_1_1_2_12"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="abian.html#D5" target="_self" title="ABIAN:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<span class="default"><a href="abian.html#K2" title="ABIAN:func.2">=_</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span><span class="kw"> such that </span><br/><a NAME="E4:39"/><span class="lab"><font color="Green" title="E16">A107</font></span>: 
 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#K8" title="EQREL_1:func.8">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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E15">A2</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c4">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#K8" title="EQREL_1:func.8">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/>
<span class="kw">set </span><font color="Maroon" title="c5">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#K8" title="EQREL_1:func.8">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/>
<span class="kw">set </span><font color="Maroon" title="c6">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#K8" title="EQREL_1:func.8">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/>
<span class="kw">set </span><font color="Maroon" title="c7">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#K8" title="EQREL_1:func.8">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/>
<span class="kw">set </span><font color="Maroon" title="c8">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#K8" title="EQREL_1:func.8">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/>
<span class="kw">set </span><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#K8" title="EQREL_1:func.8">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/>
<span class="kw">take </span>
 <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#K8" title="EQREL_1:func.8">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> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">E2</font>, <font color="Olive" title="b2">E3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <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="b3">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span> where <font color="Olive" title="b3">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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> <font color="Olive" title="b1">E2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">E3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &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"><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="b3">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span> where <font color="Olive" title="b3">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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="b3">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span> where <font color="Olive" title="b3">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">E2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">E2</font> &amp; <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b2">E3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">E3</font> )</span><br/>

<span class="kw">take </span>
 <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#K8" title="EQREL_1:func.8">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> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">E3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <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="b2">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span> where <font color="Olive" title="b2">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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="b2">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span> where <font color="Olive" title="b2">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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> <font color="Olive" title="b1">E3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &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"><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="b2">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span> where <font color="Olive" title="b2">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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="b2">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a></span>)</span> where <font color="Olive" title="b2">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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="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="b2">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span> where <font color="Olive" title="b2">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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="b2">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a></span>)</span> where <font color="Olive" title="b2">c</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">E3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">E3</font> )</span><br/>

<span class="kw">take </span>
 <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#K8" title="EQREL_1:func.8">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> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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> &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"><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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="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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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="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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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><br/>

<span class="kw">thus </span><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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="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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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="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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><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> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> )</span><br/><span class="kw">assume </span><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font></span><br/><a NAME="E2:39_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A108</font></span>: 
( <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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_2_1"><span class="lab"><font color="Green" title="E17">A108</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1"><span class="kw">suppose </span></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#K8" title="EQREL_1:func.8">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_2_1_1_1"/><span class="lab"><font color="Green" title="E18">A109</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:39_2_1_1_1"/><span class="lab"><font color="Green" title="E19">A110</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E5:39_2_1_1_1"/>
 ex <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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> st <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="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_2_1_1_1"><span class="lab"><font color="Green" title="E19">A110</font></span></a></span>;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_2_1_1_1"><span class="lab"><font color="Green" title="E18">A109</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_2"><span class="kw">suppose </span></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#K8" title="EQREL_1:func.8">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_2_1_1_2"/><span class="lab"><font color="Green" title="E18">A111</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:39_2_1_1_2"/><span class="lab"><font color="Green" title="E19">A112</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E5:39_2_1_1_2"/>
 ex <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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> st <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="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_2_1_1_2"><span class="lab"><font color="Green" title="E19">A112</font></span></a></span>;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_2_1_1_2"><span class="lab"><font color="Green" title="E18">A111</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_3"><span class="kw">suppose </span></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#K8" title="EQREL_1:func.8">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_2_1_1_3"/><span class="lab"><font color="Green" title="E18">A113</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:39_2_1_1_3"/><span class="lab"><font color="Green" title="E19">A114</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E5:39_2_1_1_3"/>
 ex <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">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> st <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="Olive" title="b1">c</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_2_1_1_3"><span class="lab"><font color="Green" title="E19">A114</font></span></a></span>;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_2_1_1_3"><span class="lab"><font color="Green" title="E18">A113</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> 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"><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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><br/>

<span class="kw">set </span><font color="Maroon" title="c11">c</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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/>
<span class="kw">assume </span><a NAME="E2:39_2"/><span class="lab"><font color="Green" title="E17">A115</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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><br/>

<a NAME="E3:39_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A116</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T20" title="EQREL_1:th.20">EQREL_1:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">c</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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#K8" title="EQREL_1:func.8">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="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_2"><span class="lab"><font color="Green" title="E17">A115</font></span></a>, <a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E16">A107</font></span></a>, <a class="txt" href="#E3:39_2"><span class="lab"><font color="Green" title="E18">A116</font></span></a></span>;<br/>
<a NAME="E6:39_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A117</font></span>: 
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_2"><span class="lab"><font color="Green" title="E19">A117</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:39_2_2_1"/><span class="lab"><font color="Green" title="E20">A118</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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><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#K8" title="EQREL_1:func.8">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"/><span class="kw">then </span>
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2_2_1"><span class="lab"><font color="Green" title="E20">A118</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E4:39_2_2_1"/><span class="kw">then </span>
<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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:39_2_2_2"/><span class="lab"><font color="Green" title="E20">A119</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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><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#K8" title="EQREL_1:func.8">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"/><span class="kw">then </span>
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2_2_2"><span class="lab"><font color="Green" title="E20">A119</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E4:39_2_2_2"/><span class="kw">then </span>
<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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_2_3"><span class="kw">suppose </span></a><a NAME="E1:39_2_2_3"/><span class="lab"><font color="Green" title="E20">A120</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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><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#K8" title="EQREL_1:func.8">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"/><span class="kw">then </span>
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2_2_3"><span class="lab"><font color="Green" title="E20">A120</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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="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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_3"/><span class="lab"><font color="Green" title="E17">A121</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">and </span><br/><a NAME="E4:39_3"/><span class="lab"><font color="Green" title="E18">A122</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:39_3"/><span class="lab"><font color="Green" title="E19">A123</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:39_3"/><span class="lab"><font color="Green" title="E20">A124</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_3"><span class="lab"><font color="Green" title="E18">A122</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><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#K8" title="EQREL_1:func.8">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="kw"> such that </span><br/><a NAME="E9:39_3"/><span class="lab"><font color="Green" title="E21">A125</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_3"><span class="lab"><font color="Green" title="E20">A124</font></span></a></span>;<br/>
<a NAME="E10:39_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_3"><span class="lab"><font color="Green" title="E19">A123</font></span></a>, <a class="txt" href="#E9:39_3"><span class="lab"><font color="Green" title="E21">A125</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E11:39_3"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E12:39_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A126</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E16">A107</font></span></a></span>;<br/>
<a NAME="E13:39_3"/>
 ex <font color="Olive" title="b1">x9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x9</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#K6" title="EQREL_1:func.6">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="Olive" title="b1">x9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/>
<a NAME="E14:39_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A127</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39_3"><span class="lab"><font color="Green" title="E22">A126</font></span></a>, <a class="ref" href="eqrel_1.html#T23" title="EQREL_1:th.23">EQREL_1:23</a></span>;<br/>
<a NAME="E15:39_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E16:39_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A128</font></span>: 
<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#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_3"><span class="lab"><font color="Green" title="E19">A123</font></span></a>, <a class="txt" href="#E9:39_3"><span class="lab"><font color="Green" title="E21">A125</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E18:39_3"/><span class="lab"><font color="Green" title="E25">A129</font></span>: 
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E19:39_3"/><span class="lab"><font color="Green" title="E26">A130</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">and </span><br/><a NAME="E20:39_3"/><span class="lab"><font color="Green" title="E27">A131</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_3"><span class="lab"><font color="Green" title="E17">A121</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">YY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E22:39_3"/><span class="lab"><font color="Green" title="E28">A132</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E23:39_3"/><span class="lab"><font color="Green" title="E29">A133</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_3"><span class="lab"><font color="Green" title="E26">A130</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><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#K8" title="EQREL_1:func.8">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="kw"> such that </span><br/><a NAME="E25:39_3"/><span class="lab"><font color="Green" title="E30">A134</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:39_3"><span class="lab"><font color="Green" title="E29">A133</font></span></a></span>;<br/>
<a NAME="E26:39_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:39_3"><span class="lab"><font color="Green" title="E28">A132</font></span></a>, <a class="txt" href="#E25:39_3"><span class="lab"><font color="Green" title="E30">A134</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E27:39_3"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E28:39_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A135</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E16">A107</font></span></a></span>;<br/>
<a NAME="E29:39_3"/>
 ex <font color="Olive" title="b1">xx9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">xx9</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#K6" title="EQREL_1:func.6">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="Olive" title="b1">xx9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/>
<a NAME="E30:39_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A136</font></span>: 
<font color="Maroon" title="c15">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:39_3"><span class="lab"><font color="Green" title="E31">A135</font></span></a>, <a class="ref" href="eqrel_1.html#T23" title="EQREL_1:th.23">EQREL_1:23</a></span>;<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_3"><span class="lab"><font color="Green" title="E27">A131</font></span></a>, <a class="ref" href="funct_7.html#T70" title="FUNCT_7:th.70">FUNCT_7:70</a></span>
<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"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:39_3"><span class="lab"><font color="Green" title="E24">A128</font></span></a>, <a class="ref" href="funct_1.html#T17" title="FUNCT_1:th.17">FUNCT_1:17</a></span>
<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T68" title="FUNCT_7:th.68">FUNCT_7:68</a></span>
;<br/>
<a NAME="E32:39_3"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_3"><span class="lab"><font color="Green" title="E19">A123</font></span></a>, <a class="txt" href="#E9:39_3"><span class="lab"><font color="Green" title="E21">A125</font></span></a>, <a class="txt" href="#E22:39_3"><span class="lab"><font color="Green" title="E28">A132</font></span></a>, <a class="txt" href="#E25:39_3"><span class="lab"><font color="Green" title="E30">A134</font></span></a>, <a class="ref" href="abian.html#D7" target="_self" title="ABIAN:def.7">Def7</a></span>;<br/>
<a NAME="E33:39_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A137</font></span>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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#K6" title="EQREL_1:func.6">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_3"><span class="lab"><font color="Green" title="E19">A123</font></span></a>, <a class="txt" href="#E9:39_3"><span class="lab"><font color="Green" title="E21">A125</font></span></a>, <a class="ref" href="eqrel_1.html#T35" title="EQREL_1:th.35">EQREL_1:35</a></span>;<br/>
<a NAME="E34:39_3"/><span class="lab"><font color="Green" title="E34">A138</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_3"><span class="lab"><font color="Green" title="E25">A129</font></span></a>, <a class="txt" href="#E22:39_3"><span class="lab"><font color="Green" title="E28">A132</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E35:39_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E16">A107</font></span></a>, <a class="txt" href="#E25:39_3"><span class="lab"><font color="Green" title="E30">A134</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:39_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_3"><span class="lab"><font color="Green" title="E19">A123</font></span></a>, <a class="txt" href="#E9:39_3"><span class="lab"><font color="Green" title="E21">A125</font></span></a>, <a class="txt" href="#E20:39_3"><span class="lab"><font color="Green" title="E27">A131</font></span></a>, <a class="txt" href="#E25:39_3"><span class="lab"><font color="Green" title="E30">A134</font></span></a>, <a class="txt" href="#E14:39_3"><span class="lab"><font color="Green" title="E23">A127</font></span></a>, <a class="txt" href="#E30:39_3"><span class="lab"><font color="Green" title="E32">A136</font></span></a>, <a class="txt" href="#E33:39_3"><span class="lab"><font color="Green" title="E33">A137</font></span></a>, <a class="txt" href="#E34:39_3"><span class="lab"><font color="Green" title="E34">A138</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_4"/><span class="lab"><font color="Green" title="E17">A139</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">and </span><br/><a NAME="E4:39_4"/><span class="lab"><font color="Green" title="E18">A140</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:39_4"/><span class="lab"><font color="Green" title="E19">A141</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:39_4"/><span class="lab"><font color="Green" title="E20">A142</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_4"><span class="lab"><font color="Green" title="E18">A140</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><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#K8" title="EQREL_1:func.8">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="kw"> such that </span><br/><a NAME="E9:39_4"/><span class="lab"><font color="Green" title="E21">A143</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_4"><span class="lab"><font color="Green" title="E20">A142</font></span></a></span>;<br/>
<a NAME="E10:39_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_4"><span class="lab"><font color="Green" title="E19">A141</font></span></a>, <a class="txt" href="#E9:39_4"><span class="lab"><font color="Green" title="E21">A143</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E11:39_4"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E12:39_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A144</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E16">A107</font></span></a></span>;<br/>
<a NAME="E13:39_4"/>
 ex <font color="Olive" title="b1">x9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x9</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#K6" title="EQREL_1:func.6">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="Olive" title="b1">x9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/>
<a NAME="E14:39_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A145</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39_4"><span class="lab"><font color="Green" title="E22">A144</font></span></a>, <a class="ref" href="eqrel_1.html#T23" title="EQREL_1:th.23">EQREL_1:23</a></span>;<br/>
<a NAME="E15:39_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E16:39_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A146</font></span>: 
<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#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_4"><span class="lab"><font color="Green" title="E19">A141</font></span></a>, <a class="txt" href="#E9:39_4"><span class="lab"><font color="Green" title="E21">A143</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E18:39_4"/><span class="lab"><font color="Green" title="E25">A147</font></span>: 
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E19:39_4"/><span class="lab"><font color="Green" title="E26">A148</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">and </span><br/><a NAME="E20:39_4"/><span class="lab"><font color="Green" title="E27">A149</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_4"><span class="lab"><font color="Green" title="E17">A139</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">YY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E22:39_4"/><span class="lab"><font color="Green" title="E28">A150</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E23:39_4"/><span class="lab"><font color="Green" title="E29">A151</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_4"><span class="lab"><font color="Green" title="E26">A148</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><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#K8" title="EQREL_1:func.8">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="kw"> such that </span><br/><a NAME="E25:39_4"/><span class="lab"><font color="Green" title="E30">A152</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:39_4"><span class="lab"><font color="Green" title="E29">A151</font></span></a></span>;<br/>
<a NAME="E26:39_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:39_4"><span class="lab"><font color="Green" title="E28">A150</font></span></a>, <a class="txt" href="#E25:39_4"><span class="lab"><font color="Green" title="E30">A152</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E27:39_4"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E28:39_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A153</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E16">A107</font></span></a></span>;<br/>
<a NAME="E29:39_4"/>
 ex <font color="Olive" title="b1">xx9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">xx9</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#K6" title="EQREL_1:func.6">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="Olive" title="b1">xx9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/>
<a NAME="E30:39_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A154</font></span>: 
<font color="Maroon" title="c15">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:39_4"><span class="lab"><font color="Green" title="E31">A153</font></span></a>, <a class="ref" href="eqrel_1.html#T23" title="EQREL_1:th.23">EQREL_1:23</a></span>;<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_4"><span class="lab"><font color="Green" title="E27">A149</font></span></a>, <a class="ref" href="funct_7.html#T70" title="FUNCT_7:th.70">FUNCT_7:70</a></span>
<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"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:39_4"><span class="lab"><font color="Green" title="E24">A146</font></span></a>, <a class="ref" href="funct_1.html#T17" title="FUNCT_1:th.17">FUNCT_1:17</a></span>
<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T68" title="FUNCT_7:th.68">FUNCT_7:68</a></span>
;<br/>
<a NAME="E32:39_4"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_4"><span class="lab"><font color="Green" title="E19">A141</font></span></a>, <a class="txt" href="#E9:39_4"><span class="lab"><font color="Green" title="E21">A143</font></span></a>, <a class="txt" href="#E22:39_4"><span class="lab"><font color="Green" title="E28">A150</font></span></a>, <a class="txt" href="#E25:39_4"><span class="lab"><font color="Green" title="E30">A152</font></span></a>, <a class="ref" href="abian.html#D7" target="_self" title="ABIAN:def.7">Def7</a></span>;<br/>
<a NAME="E33:39_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A155</font></span>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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#K6" title="EQREL_1:func.6">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_4"><span class="lab"><font color="Green" title="E19">A141</font></span></a>, <a class="txt" href="#E9:39_4"><span class="lab"><font color="Green" title="E21">A143</font></span></a>, <a class="ref" href="eqrel_1.html#T35" title="EQREL_1:th.35">EQREL_1:35</a></span>;<br/>
<a NAME="E34:39_4"/><span class="lab"><font color="Green" title="E34">A156</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_4"><span class="lab"><font color="Green" title="E25">A147</font></span></a>, <a class="txt" href="#E22:39_4"><span class="lab"><font color="Green" title="E28">A150</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E35:39_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E16">A107</font></span></a>, <a class="txt" href="#E25:39_4"><span class="lab"><font color="Green" title="E30">A152</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:39_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_4"><span class="lab"><font color="Green" title="E19">A141</font></span></a>, <a class="txt" href="#E9:39_4"><span class="lab"><font color="Green" title="E21">A143</font></span></a>, <a class="txt" href="#E20:39_4"><span class="lab"><font color="Green" title="E27">A149</font></span></a>, <a class="txt" href="#E25:39_4"><span class="lab"><font color="Green" title="E30">A152</font></span></a>, <a class="txt" href="#E14:39_4"><span class="lab"><font color="Green" title="E23">A145</font></span></a>, <a class="txt" href="#E30:39_4"><span class="lab"><font color="Green" title="E32">A154</font></span></a>, <a class="txt" href="#E33:39_4"><span class="lab"><font color="Green" title="E33">A155</font></span></a>, <a class="txt" href="#E34:39_4"><span class="lab"><font color="Green" title="E34">A156</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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#K8" title="EQREL_1:func.8">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#K8" title="EQREL_1:func.8">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_5"/><span class="lab"><font color="Green" title="E17">A157</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">and </span><br/><a NAME="E4:39_5"/><span class="lab"><font color="Green" title="E18">A158</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:39_5"/><span class="lab"><font color="Green" title="E19">A159</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:39_5"/><span class="lab"><font color="Green" title="E20">A160</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_5"><span class="lab"><font color="Green" title="E18">A158</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><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#K8" title="EQREL_1:func.8">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="kw"> such that </span><br/><a NAME="E9:39_5"/><span class="lab"><font color="Green" title="E21">A161</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_5"><span class="lab"><font color="Green" title="E20">A160</font></span></a></span>;<br/>
<a NAME="E10:39_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_5"><span class="lab"><font color="Green" title="E19">A159</font></span></a>, <a class="txt" href="#E9:39_5"><span class="lab"><font color="Green" title="E21">A161</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E11:39_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A162</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E16">A107</font></span></a></span>;<br/>
<a NAME="E12:39_5"/>
 ex <font color="Olive" title="b1">x9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x9</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#K6" title="EQREL_1:func.6">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="Olive" title="b1">x9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/>
<a NAME="E13:39_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A163</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39_5"><span class="lab"><font color="Green" title="E22">A162</font></span></a>, <a class="ref" href="eqrel_1.html#T23" title="EQREL_1:th.23">EQREL_1:23</a></span>;<br/>
<a NAME="E14:39_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E15:39_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A164</font></span>: 
<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#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_5"><span class="lab"><font color="Green" title="E19">A159</font></span></a>, <a class="txt" href="#E9:39_5"><span class="lab"><font color="Green" title="E21">A161</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:39_5"/><span class="lab"><font color="Green" title="E25">A165</font></span>: 
<font color="Maroon" title="c13">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E18:39_5"/><span class="lab"><font color="Green" title="E26">A166</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">and </span><br/><a NAME="E19:39_5"/><span class="lab"><font color="Green" title="E27">A167</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_5"><span class="lab"><font color="Green" title="E17">A157</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">YY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E21:39_5"/><span class="lab"><font color="Green" title="E28">A168</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E22:39_5"/><span class="lab"><font color="Green" title="E29">A169</font></span>: 
<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#K8" title="EQREL_1:func.8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_5"><span class="lab"><font color="Green" title="E26">A166</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><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#K8" title="EQREL_1:func.8">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="kw"> such that </span><br/><a NAME="E24:39_5"/><span class="lab"><font color="Green" title="E30">A170</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:39_5"><span class="lab"><font color="Green" title="E29">A169</font></span></a></span>;<br/>
<a NAME="E25:39_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:39_5"><span class="lab"><font color="Green" title="E28">A168</font></span></a>, <a class="txt" href="#E24:39_5"><span class="lab"><font color="Green" title="E30">A170</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E26:39_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A171</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E16">A107</font></span></a></span>;<br/>
<a NAME="E27:39_5"/>
 ex <font color="Olive" title="b1">xx9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">xx9</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#K6" title="EQREL_1:func.6">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="Olive" title="b1">xx9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/>
<a NAME="E28:39_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A172</font></span>: 
<font color="Maroon" title="c15">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:39_5"><span class="lab"><font color="Green" title="E31">A171</font></span></a>, <a class="ref" href="eqrel_1.html#T23" title="EQREL_1:th.23">EQREL_1:23</a></span>;<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_5"><span class="lab"><font color="Green" title="E27">A167</font></span></a>, <a class="ref" href="funct_7.html#T70" title="FUNCT_7:th.70">FUNCT_7:70</a></span>
<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"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39_5"><span class="lab"><font color="Green" title="E24">A164</font></span></a>, <a class="ref" href="funct_1.html#T17" title="FUNCT_1:th.17">FUNCT_1:17</a></span>
<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T68" title="FUNCT_7:th.68">FUNCT_7:68</a></span>
;<br/>
<a NAME="E30:39_5"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_5"><span class="lab"><font color="Green" title="E19">A159</font></span></a>, <a class="txt" href="#E9:39_5"><span class="lab"><font color="Green" title="E21">A161</font></span></a>, <a class="txt" href="#E21:39_5"><span class="lab"><font color="Green" title="E28">A168</font></span></a>, <a class="txt" href="#E24:39_5"><span class="lab"><font color="Green" title="E30">A170</font></span></a>, <a class="ref" href="abian.html#D7" target="_self" title="ABIAN:def.7">Def7</a></span>;<br/>
<a NAME="E31:39_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A173</font></span>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">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#K6" title="EQREL_1:func.6">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_5"><span class="lab"><font color="Green" title="E19">A159</font></span></a>, <a class="txt" href="#E9:39_5"><span class="lab"><font color="Green" title="E21">A161</font></span></a>, <a class="ref" href="eqrel_1.html#T35" title="EQREL_1:th.35">EQREL_1:35</a></span>;<br/>
<a NAME="E32:39_5"/><span class="lab"><font color="Green" title="E34">A174</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:39_5"><span class="lab"><font color="Green" title="E25">A165</font></span></a>, <a class="txt" href="#E21:39_5"><span class="lab"><font color="Green" title="E28">A168</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E33:39_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E16">A107</font></span></a>, <a class="txt" href="#E24:39_5"><span class="lab"><font color="Green" title="E30">A170</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E34:39_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_5"><span class="lab"><font color="Green" title="E19">A159</font></span></a>, <a class="txt" href="#E9:39_5"><span class="lab"><font color="Green" title="E21">A161</font></span></a>, <a class="txt" href="#E19:39_5"><span class="lab"><font color="Green" title="E27">A167</font></span></a>, <a class="txt" href="#E24:39_5"><span class="lab"><font color="Green" title="E30">A170</font></span></a>, <a class="txt" href="#E13:39_5"><span class="lab"><font color="Green" title="E23">A163</font></span></a>, <a class="txt" href="#E28:39_5"><span class="lab"><font color="Green" title="E32">A172</font></span></a>, <a class="txt" href="#E31:39_5"><span class="lab"><font color="Green" title="E33">A173</font></span></a>, <a class="txt" href="#E32:39_5"><span class="lab"><font color="Green" title="E34">A174</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
