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

<b>set </b><font color="Maroon" title="c1">Z</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi'</font>) where <font color="Olive" title="b1">phi'</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi'</font>) where <font color="Olive" title="b1">phi'</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span> ;<br/>
<b>assume </b><a NAME="E2:41_1"/><i><font color="Green" title="E14">A4</font></i>: 
not <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi'</font>) where <font color="Olive" title="b1">phi'</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:41_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi'</font>) where <font color="Olive" title="b1">phi'</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E4:41_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi'</font>) where <font color="Olive" title="b1">phi'</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c3">phi</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E6:41_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c3">phi</font>)
 <b>and </b><br/><a NAME="E7:41_1"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c3">phi</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><a NAME="E1:41_1_1"/><i><font color="Green" title="E16">A6</font></i>: 
( <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">phi</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">phi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>assume </b><a NAME="E2:41_1_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><a NAME="E3:41_1_1"/><b>then </b>
<font color="Maroon" title="c3">phi</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="fraenkel.html#E7:41_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/><a NAME="E4:41_1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:41_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="relat_1.html#T151" title="RELAT_1:th.151">RELAT_1:151</a></i>;<br/><b>hence </b><a NAME="E5:41_1_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">FF</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>, <font color="Maroon" title="c5">Z</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi'</font>) where <font color="Olive" title="b1">phi'</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span>  as  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>  <b>by </b><i><a class="txt" href="fraenkel.html#E2:41_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E10:41_1"/><i><font color="Green" title="E16">A7</font></i>: 
( <font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:41_1"/><i><font color="Green" title="E17">A8</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">phi</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive" title="b1">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi</font>);<br/>
<div><i><font color="Green" title="E18">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">FF</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Z</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">g</font>]</span><br/><b>consider </b><font color="Maroon" title="c7">phi</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E2:41_1_2"/><i><font color="Green" title="E19">A10</font></i>: 
<font color="Maroon" title="c7">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="fraenkel.html#E10:41_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="fraenkel.html#T14" target="_self" title="FRAENKEL:th.14">Th14</a></i>;<br/><a NAME="E3:41_1_2"/><i><font color="Green" title="E20">A11</font></i>: 
<font color="Maroon" title="c7">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:41_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="fraenkel.html#T15" target="_self" title="FRAENKEL:th.15">Th15</a></i>;<br/><a NAME="E4:41_1_2"/>
 ex <font color="Olive" title="b1">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>() &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>() )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E5:41_1_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i><a class="txt" href="fraenkel.html#E11:41_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="fraenkel.html#E3:41_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:41_1_2"/><b>then </b>
<font color="Maroon" title="c7">phi</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/><a NAME="E7:41_1_2"/><b>then </b>
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c7">phi</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">g'</font> = <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c7">phi</font>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Z</font> ;<br/><b>take </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c9">g</font>]</span><br/><b>thus </b><a NAME="E9:41_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c9">g</font>]
 <b>by </b><i><a class="txt" href="fraenkel.html#E3:41"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="fraenkel.html#E3:41_1_2"><i><font color="Green" title="E20">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">FF</font>,<font color="Maroon" title="c5">Z</font><b> such that </b><br/><a NAME="E14:41_1"/><i><font color="Green" title="E19">A12</font></i>: 
for <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">FF</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="fraenkel.html#E12:41_1"><i><font color="Green" title="E18">A9</font></i></a>);<br/></i>
<a NAME="E15:41_1"/>
<font color="Maroon" title="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> or <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:41_1_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c8">phi</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:41_1_3"/><i><font color="Green" title="E20">A13</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c8">phi</font>)
 <b>and </b><br/><a NAME="E4:41_1_3"/><i><font color="Green" title="E21">A14</font></i>: 
<font color="Maroon" title="c8">phi</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 ;<br/>
<i><font color="Green" title="E22">A15</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">phi</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon">F<sub>3</sub></font>()
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E6:41_1_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 
<b>by </b><i><a class="txt" href="fraenkel.html#E4:41_1_3"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/>
<a NAME="E7:41_1_3"/><b>then </b><i><font color="Green" title="E23">A16</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E8:41_1_3"/><b>then </b><i><font color="Green" title="E24">A17</font></i>: 
<font color="Maroon" title="c8">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="fraenkel.html#E5:41_1_3"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>() as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <b>by </b><i><a class="txt" href="fraenkel.html#E5:41_1_3"><i><font color="Green" title="E22">A15</font></i></a>, <a class="txt" href="fraenkel.html#E7:41_1_3"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E10:41_1_3"/><i><font color="Green" title="E25">A18</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 
<b>by </b><i><a class="txt" href="fraenkel.html#E8:41_1_3"><i><font color="Green" title="E24">A17</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:41_1_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="txt" href="fraenkel.html#E14:41_1"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="fraenkel.html#E3:41_1_3"><i><font color="Green" title="E20">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:41_1_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fraenkel.html#E10:41_1_3"><i><font color="Green" title="E25">A18</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E16:41_1"/><b>then </b><i><font color="Green" title="E20">A19</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E17:41_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>() is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>() is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="fraenkel.html#E1:41"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="fraenkel.html#E2:41"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="finset_1.html#T15" title="FINSET_1:th.15">FINSET_1:15</a></i>;<br/>
<a NAME="E18:41_1"/><b>then </b>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="fraenkel.html#T16" target="_self" title="FRAENKEL:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E19:41_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:41_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="fraenkel.html#E16:41_1"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="finset_1.html#T17" title="FINSET_1:th.17">FINSET_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
