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

<b>set </b><font color="Maroon" title="c1">F1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c2">F2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> ;<br/>
<a NAME="E1:41_1"/><i><font color="Green" title="E28">A5</font></i>: 
( <font color="Maroon">F<sub>4</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies <font color="Maroon">F<sub>3</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon">F<sub>4</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>assume </b><a NAME="E2:41_1_1"/>
not <font color="Maroon">F<sub>3</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:41_1_1"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>thus </b><a NAME="E5:41_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E2:41"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="stirl2_1.html#E3:41"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="stirl2_1.html#E4:41_1_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div>
<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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<br/> for <font color="Olive" title="b2">h</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/><font color="Olive" title="b2">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>;<br/>
<a NAME="E2:41_1"/><i><font color="Green" title="E29">A8</font></i>: 
for <font color="Olive" title="b1">f'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  holds <br/> ex <font color="Olive" title="b2">g'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">g'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">f</font> where <font color="Olive" title="b3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b4">x</font>) ) ) </span>}</span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f'</font>,<font color="Olive" title="b2">g'</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">f'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:41_1_2"/><i><font color="Green" title="E30">A9</font></i>: 
<font color="Maroon" title="c3">f'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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_2"/><i><font color="Green" title="E31">A10</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f'</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_2"><i><font color="Green" title="E30">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<b> such that </b><br/><a NAME="E5:41_1_2"/><i><font color="Green" title="E32">A11</font></i>: 
( <font color="Maroon" title="c5">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S2" target="_self" title="STIRL2_1:sch.2">STIRL2_1:sch 2</a>(<a class="txt" href="stirl2_1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E2:41"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="stirl2_1.html#E3:41"><i><font color="Green" title="E26">A3</font></i></a>);<br/></i>
<a NAME="E6:41_1_2"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">h</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</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="txt" href="stirl2_1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="stirl2_1.html#E3:41_1_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="txt" href="stirl2_1.html#E5:41_1_2"><i><font color="Green" title="E32">A11</font></i></a></i>;<br/>
<a NAME="E7:41_1_2"/><b>then </b>
( <font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f'</font>,<font color="Maroon" title="c5">h</font>] )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E3:41_1_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="txt" href="stirl2_1.html#E5:41_1_2"><i><font color="Green" title="E32">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:41_1_2"/>
 ex <font color="Olive" title="b1">g'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">g'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b3">x</font>) ) ) </span>}</span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f'</font>,<font color="Olive" title="b1">g'</font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">ff</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span> ,<span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> <b> such that </b><br/><a NAME="E4:41_1"/><i><font color="Green" title="E30">A12</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="stirl2_1.html#E2:41_1"><i><font color="Green" title="E29">A8</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_3_1"/>
( <font color="Maroon">F<sub>4</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon">F<sub>4</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_1"><b>suppose </b></a><a NAME="E1:41_1_3_1_1"/><i><font color="Green" title="E31">A13</font></i>: 
<font color="Maroon">F<sub>4</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:41_1_3_1_1"/><i><font color="Green" title="E32">A14</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_3_1_1_1"/><i><font color="Green" title="E33">A15</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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_1_1_1"/><i><font color="Green" title="E34">A16</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_1_1"><i><font color="Green" title="E33">A15</font></i></a></i>;<br/>
<a NAME="E4:41_1_3_1_1_1"/>
<font color="Maroon">F<sub>2</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:41"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1_3_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E5:41_1_3_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</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_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E6:41_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</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="stirl2_1.html#E3:41_1_3_1_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<b>hence </b><a NAME="E7:41_1_3_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:41_1_3_1_1"/><i><font color="Green" title="E33">A17</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_3_1_1_2"/><i><font color="Green" title="E34">A18</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<b> such that </b><br/><a NAME="E3:41_1_3_1_1_2"/><i><font color="Green" title="E35">A19</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_1_2"><i><font color="Green" title="E34">A18</font></i></a></i>;<br/>
<a NAME="E4:41_1_3_1_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</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="stirl2_1.html#E1:41_1_3_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E5:41_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">x</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="stirl2_1.html#E3:41_1_3_1_1_2"><i><font color="Green" title="E35">A19</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<b>hence </b><a NAME="E6:41_1_3_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c4">Empty</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_3_1_1_3_1"/>
( <font color="Maroon">P<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ] or  not <font color="Maroon">P<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:41_1_3_1_1_3_1_1"/><i><font color="Green" title="E34">A20</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 ;<br/><div class="add"><a NAME="E2:41_1_3_1_1_3_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="stirl2_1.html#E2:41"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1_3_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">Empty</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1_3_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/><a NAME="E4:41_1_3_1_1_3_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:41"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1_3_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><a NAME="E5:41_1_3_1_1_3_1_1"/><b>then </b>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Empty</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c5">Empty</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:41"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1_3_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/><a NAME="E6:41_1_3_1_1_3_1_1"/><b>then </b>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1_3_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1_3_1_1_3_1_1"><i><font color="Green" title="E34">A20</font></i></a></i>;<br/><a NAME="E7:41_1_3_1_1_3_1_1"/><b>then </b>
( <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:41_1_3_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="stirl2_1.html#E3:41_1_3_1_1"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/><b>hence </b><a NAME="E8:41_1_3_1_1_3_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:41_1_3_1_1_3_1_2"/><i><font color="Green" title="E34">A21</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 ;<br/><div class="add"><a NAME="E2:41_1_3_1_1_3_1_2"/><i><font color="Green" title="E35">A22</font></i>: 
not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1_1_3_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_3_1_1_3_1_2_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span> 
 ;<br/>

<a NAME="E2:41_1_3_1_1_3_1_2_1"/><b>then </b>
( <font color="Maroon">F<sub>2</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon">F<sub>1</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  ex <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">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] ) )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:41"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="stirl2_1.html#E3:41"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1_3_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_3_1_1_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_1_3_1_2"><i><font color="Green" title="E34">A21</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:41_1_3_1_1_3_1_2"/><i><font color="Green" title="E36">A23</font></i>: 
not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1_1_3_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_3_1_1_3_1_2_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 ;<br/>

<a NAME="E2:41_1_3_1_1_3_1_2_2"/><b>then </b>
( <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>  &amp; <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>  &amp;  ex <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">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1_3_1_1"><i><font color="Green" title="E31">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_3_1_1_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_1_3_1_2"><i><font color="Green" title="E34">A21</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:41_1_3_1_1_3_1_2"/>
( ( <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> ) &amp; ( <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> ) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:41_1_3_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="stirl2_1.html#E3:41_1_3_1_1"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/><b>hence </b><a NAME="E5:41_1_3_1_1_3_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:41_1_3_1_1_3_1_2"><i><font color="Green" title="E35">A22</font></i></a>, <a class="txt" href="stirl2_1.html#E3:41_1_3_1_1_3_1_2"><i><font color="Green" title="E36">A23</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:41_1_3_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_2"><b>suppose </b></a><a NAME="E1:41_1_3_1_2"/><i><font color="Green" title="E31">A24</font></i>: 
not <font color="Maroon">F<sub>4</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_3_1_2_1_1"/>
( <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_3_1_2_1_1_1"/><i><font color="Green" title="E32">A25</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:41_1_3_1_2_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_3_1_2_1_1_1_1"/>
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:41_1_3_1_2_1_1_1_1"/><i><font color="Green" title="E33">A26</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E5:41_1_3_1_2_1_1_1_1"/><i><font color="Green" title="E34">A27</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:41_1_3_1_2_1_1_1_1"><i><font color="Green" title="E33">A26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<b> such that </b><br/><a NAME="E7:41_1_3_1_2_1_1_1_1"/><i><font color="Green" title="E35">A28</font></i>: 
( <font color="Maroon" title="c6">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S2" target="_self" title="STIRL2_1:sch.2">STIRL2_1:sch 2</a>(<a class="txt" href="stirl2_1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E2:41"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="stirl2_1.html#E3:41"><i><font color="Green" title="E26">A3</font></i></a>);<br/></i>
<a NAME="E8:41_1_3_1_2_1_1_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">h</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</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="txt" href="stirl2_1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="stirl2_1.html#E5:41_1_3_1_2_1_1_1_1"><i><font color="Green" title="E34">A27</font></i></a>, <a class="txt" href="stirl2_1.html#E7:41_1_3_1_2_1_1_1_1"><i><font color="Green" title="E35">A28</font></i></a></i>;<br/>
<a NAME="E9:41_1_3_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E7:41_1_3_1_2_1_1_1_1"><i><font color="Green" title="E35">A28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:41_1_3_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_1"><i><font color="Green" title="E32">A25</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:41_1_3_1_2_1_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_1"><i><font color="Green" title="E32">A25</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_3_1_2_1_1_2"/><i><font color="Green" title="E32">A29</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:41_1_3_1_2_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">ff</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_3_1_2_1_1_2_1"/><i><font color="Green" title="E33">A30</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<b> such that </b><br/><a NAME="E3:41_1_3_1_2_1_1_2_1"/><i><font color="Green" title="E34">A31</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_2_1"><i><font color="Green" title="E33">A30</font></i></a></i>;<br/>
<a NAME="E4:41_1_3_1_2_1_1_2_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:41_1_3_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E35">A32</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="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:41"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">h</font> = <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/>
<a NAME="E7:41_1_3_1_2_1_1_2_1"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies <font color="Maroon">F<sub>1</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E5:41_1_3_1_2_1_1_2_1"><i><font color="Green" title="E35">A32</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">h</font> = <font color="Maroon" title="c6">h</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="stirl2_1.html#E3:41_1_3_1_2_1_1_2_1"><i><font color="Green" title="E34">A31</font></i></a>, <a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></i>;<br/>
<a NAME="E9:41_1_3_1_2_1_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()]
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="stirl2_1.html#E3:41_1_3_1_2_1_1_2_1"><i><font color="Green" title="E34">A31</font></i></a></i>;<br/>
<a NAME="E10:41_1_3_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E36">A33</font></i>: 
( <font color="Maroon" title="c7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_2"><i><font color="Green" title="E32">A29</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_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">ff</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:41_1_3_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c8">ffh</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<b> such that </b><br/><a NAME="E14:41_1_3_1_2_1_1_2_1"/><i><font color="Green" title="E37">A34</font></i>: 
( <font color="Maroon" title="c8">ffh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">h</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c8">ffh</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">ffh</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c8">ffh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:41_1_3_1_2_1_1_2_1_1"/><i><font color="Green" title="E38">A35</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1_2_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_2_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_3_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_2_1_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_3_1_2_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><div class="add"><a NAME="E2:41_1_3_1_2_1_1_2_1_1_1_1_1"/><b>then </b>
( <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="c7">h</font> &amp; <font color="Maroon" title="c8">ffh</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:41"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="stirl2_1.html#E4:41_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="stirl2_1.html#E10:41_1_3_1_2_1_1_2_1"><i><font color="Green" title="E36">A33</font></i></a>, <a class="txt" href="stirl2_1.html#E14:41_1_3_1_2_1_1_2_1"><i><font color="Green" title="E37">A34</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="E3:41_1_3_1_2_1_1_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">ffh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</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="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E4:41_1_3_1_2_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c8">ffh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_2_1_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_3_1_2_1_1_2_1_1_1_1_2"/>
not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><div class="add"><a NAME="E2:41_1_3_1_2_1_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_2_1_1"><i><font color="Green" title="E38">A35</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:41_1_3_1_2_1_1_2_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">ffh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c9">x</font>) &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c9">x</font>) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:41_1_3_1_2_1_1_2_1"><i><font color="Green" title="E34">A31</font></i></a>, <a class="txt" href="stirl2_1.html#E14:41_1_3_1_2_1_1_2_1"><i><font color="Green" title="E37">A34</font></i></a></i>;<br/><b>hence </b><a NAME="E4:41_1_3_1_2_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c8">ffh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:41_1_3_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c8">ffh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E16:41_1_3_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">ffh</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>
<b>hence </b><a NAME="E17:41_1_3_1_2_1_1_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">ff</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:41_1_3_1_2_1_1_2_1"><i><font color="Green" title="E34">A31</font></i></a>, <a class="txt" href="stirl2_1.html#E10:41_1_3_1_2_1_1_2_1"><i><font color="Green" title="E36">A33</font></i></a>, <a class="txt" href="stirl2_1.html#E14:41_1_3_1_2_1_1_2_1"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:41_1_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E33">A36</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_2"><i><font color="Green" title="E32">A29</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E4:41_1_3_1_2_1_1_2"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">f</font> where <font color="Olive" title="b3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">f</font> where <font color="Olive" title="b3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  &amp; <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:41_1_3_1_2_1_1_2_2"/><i><font color="Green" title="E34">A37</font></i>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  &amp; <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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_1_2_1_1_2_2"/><i><font color="Green" title="E35">A38</font></i>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">f1</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_2_2"><i><font color="Green" title="E34">A37</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">f2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E5:41_1_3_1_2_1_1_2_2"/><i><font color="Green" title="E36">A39</font></i>: 
( <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">f2</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_2_2"><i><font color="Green" title="E34">A37</font></i></a></i>;<br/>
<a NAME="E6:41_1_3_1_2_1_1_2_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span> 
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_2"><i><font color="Green" title="E32">A29</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="E7:41_1_3_1_2_1_1_2_2"/><b>then </b>
( <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">ff</font> &amp; <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">ff</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_2_2"><i><font color="Green" title="E34">A37</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:41_1_3_1_2_1_1_2_2"/><b>then </b><i><font color="Green" title="E37">A40</font></i>: 
( <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  &amp; <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c8">F1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<b> such that </b><br/><a NAME="E10:41_1_3_1_2_1_1_2_2"/><i><font color="Green" title="E38">A41</font></i>: 
( <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F1</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c8">F1</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c8">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">F2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<b> such that </b><br/><a NAME="E12:41_1_3_1_2_1_1_2_2"/><i><font color="Green" title="E39">A42</font></i>: 
( <font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">F2</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c9">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E8:41_1_3_1_2_1_1_2_2"><i><font color="Green" title="E37">A40</font></i></a></i>;<br/>
<a NAME="E13:41_1_3_1_2_1_1_2_2"/>
( <font color="Maroon" title="c8">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> &amp; <font color="Maroon" title="c9">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> &amp; <font color="Maroon" title="c8">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">F2</font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E4:41_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_2_2"><i><font color="Green" title="E34">A37</font></i></a>, <a class="txt" href="stirl2_1.html#E3:41_1_3_1_2_1_1_2_2"><i><font color="Green" title="E35">A38</font></i></a>, <a class="txt" href="stirl2_1.html#E5:41_1_3_1_2_1_1_2_2"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="stirl2_1.html#E10:41_1_3_1_2_1_1_2_2"><i><font color="Green" title="E38">A41</font></i></a>, <a class="txt" href="stirl2_1.html#E12:41_1_3_1_2_1_1_2_2"><i><font color="Green" title="E39">A42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:41_1_3_1_2_1_1_2_2"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:41_1_3_1_2_1_1_2_2"><i><font color="Green" title="E35">A38</font></i></a>, <a class="txt" href="stirl2_1.html#E5:41_1_3_1_2_1_1_2_2"><i><font color="Green" title="E36">A39</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:41_1_3_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">ff</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:41_1_3_1_2_1_1_2"><i><font color="Green" title="E32">A29</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/><a NAME="E6:41_1_3_1_2_1_1_2"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span> ,<span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span>  <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:41_1_3_1_2_1_1_2"><i><font color="Green" title="E33">A36</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/><b>hence </b><a NAME="E7:41_1_3_1_2_1_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:41_1_3_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:41_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()] &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>) ) ) </span>}</span> 
 ;<br/>


</div>
