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

<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>( ( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 ) &amp; ( $1 <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>() implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>($1) ) );<br/>
<a NAME="E1:40_1"/><i><font color="Green" title="E27">A4</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> <font color="Maroon">F<sub>3</sub></font>() holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_1_1"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c1">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="40_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_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="E1:40_1_1_1_1"/>
( <font color="Maroon" title="c1">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="c1">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="40_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_1_1_1_1"/>
<font color="Maroon" title="c1">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="E2:40_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">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">F<sub>5</sub></font>()
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:40"><i><font color="Green" title="E26">A3</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="E3:40_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</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">F<sub>5</sub></font>() &amp; not <font color="Maroon" title="c1">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="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:40_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>] &amp; <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:40"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E5:40_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">y</font>] )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_1_1_2"/><i><font color="Green" title="E29">A6</font></i>: 
not <font color="Maroon" title="c1">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="E2:40_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">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#E1:40_1_1"><i><font color="Green" title="E28">A5</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="E3:40_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c1">x</font>)] &amp; <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:40"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E1:40_1_1_1_1_2"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E4:40_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">y</font>] )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:40_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">y</font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c1">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="E3:40_1"/><i><font color="Green" title="E28">A7</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> <font color="Maroon">F<sub>3</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">h</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#E1:40_1"><i><font color="Green" title="E27">A4</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c1">h</font>
;<br/>

<b>thus </b><a NAME="E4:40_1"/>
<font color="Maroon" title="c1">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">F<sub>5</sub></font>()
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:40_1_2"/><i><font color="Green" title="E29">A8</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon">F<sub>5</sub></font>() <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="c1">h</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_2_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> or  not <font color="Maroon">F<sub>2</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="40_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_2_1_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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_1_2_1_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon">F<sub>5</sub></font>() <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="c1">h</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:40"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_2_1_1_1_2"/><i><font color="Green" title="E30">A9</font></i>: 
not <font color="Maroon">F<sub>2</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:40_1_2_1_1_1_2"/><b>then </b>
not <font color="Maroon">F<sub>4</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#E2:40"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><a NAME="E3:40_1_2_1_1_1_2"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon">F<sub>5</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">h</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#E1:40_1_2_1_1_1_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E4:40_1_2_1_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon">F<sub>5</sub></font>() <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="c1">h</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:40"><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/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_1_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon">F<sub>5</sub></font>() <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="c1">h</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:40_1_2"/>
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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon">F<sub>5</sub></font>() holds <br/><font color="Maroon" title="c1">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>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_1_2_2"/><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c2">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">F<sub>5</sub></font>()
 ;<br/>

<a NAME="E2:40_1_2_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:40_1_2_2"><i><font color="Green" title="E30">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:40_1_2_2"/>
<font color="Maroon" title="c1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:40"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="stirl2_1.html#E3:40_1"><i><font color="Green" title="E28">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40_1_2"/>
<font color="Maroon" title="c1">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">F<sub>5</sub></font>()
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:40_1_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


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

<b>thus </b><a NAME="E5:40_1"/>
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="c1">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>6</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:40_1"><i><font color="Green" title="E28">A7</font></i></a></i>;<br/>


</div>
