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

<b>set </b><font color="Maroon" title="c1">D</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1"/>
( <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</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">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] ) </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</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> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:12_1_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; ( for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</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" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">u</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E4:12_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">v</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="E1:12_1_1_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_1_1_1_1"/><i><font color="Green" title="E11">A5</font></i>: 
( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</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="E4:12_1_1_1_1"/>
not <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="topgen_5.html#E1:12_1_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<a NAME="E5:12_1_1_1_1"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">u</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">u</font>] &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c4">u</font>) )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E3:12_1_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="topgen_5.html#E3:12_1_1_1_1"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:12_1_1_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="topgen_5.html#E2:12"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="topgen_5.html#E3:12_1_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="topgen_5.html#E3:12_1_1_1_1"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">f</font> 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>2</sub></font>() <b>by </b><i><a class="txt" href="topgen_5.html#E3:12_1_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c3">f</font>
;<br/><b>let </b><font color="Maroon" title="c4">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><a NAME="E6:12_1_1_1"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 ;<br/><a NAME="E7:12_1_1_1"/>
not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:12_1_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E8:12_1_1_1"/>
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>] implies <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c4">a</font>) ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">a</font>] implies <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c4">a</font>) ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">a</font>] implies <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c4">a</font>) ) )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:12_1_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="topgen_5.html#E6:12_1_1_1"><i><font color="Green" title="E11">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] ) </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c2">D</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E3:12_1_1_2"/><i><font color="Green" title="E9">A7</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> holds <br/>( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] implies <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] implies <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_1_1_2_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 ;<br/>

<a NAME="E2:12_1_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">b</font>] ) )
 
;<br/>
<a NAME="E3:12_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">a</font>] ) )
 
;<br/>
<b>hence </b><a NAME="E4:12_1_1_2_1"/>
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>] implies <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>] implies <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="topgen_5.html#E2:12"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c3">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E5:12_1_1_2"/><i><font color="Green" title="E10">A8</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" title="c2">D</font> holds <br/>( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c3">g</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>4</sub></font>(<font color="Olive" title="b1">x</font>) ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c3">g</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="funct_2.html#S5" title="FUNCT_2:sch.5">FUNCT_2:sch 5</a>(<a class="txt" href="topgen_5.html#E3:12_1_1_2"><i><font color="Green" title="E9">A7</font></i></a>);<br/></i><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[$1] or <font color="Maroon">P<sub>2</sub></font>[$1] );<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/><a NAME="E6:12_1_1_2"/><i><font color="Green" title="E11">A9</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</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">a</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] implies <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_1_1_2_2"/><i><font color="Green" title="E12">A10</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1_1_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>]
 ;<br/><a NAME="E2:12_1_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:12_1_1_2_2"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/><a NAME="E3:12_1_1_2_2_1"/><b>then </b>
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>] implies <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c4">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>] implies <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c4">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>] implies <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c4">a</font>) ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>] implies <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c4">a</font>) ) )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:12_1_1_2"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="topgen_5.html#E5:12_1_1_2"><i><font color="Green" title="E10">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E4:12_1_1_2_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E3:12_1_1_2_2"/>
(  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>] implies <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c4">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="topgen_5.html#E1:12_1_1_2_2"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="topgen_5.html#E2:12"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>


</div><b>end;</b></div><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>3</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E8:12_1_1_2"/><i><font color="Green" title="E12">A11</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>] implies <font color="Maroon" title="c4">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">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c4">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>6</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5" title="FUNCT_2:sch.5">FUNCT_2:sch 5</a>(<a class="txt" href="topgen_5.html#E6:12_1_1_2"><i><font color="Green" title="E11">A9</font></i></a>);<br/></i><b>take </b>
<font color="Maroon" title="c4">f</font>
;<br/><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><a NAME="E9:12_1_1_2"/><i><font color="Green" title="E13">A12</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 ;<br/><a NAME="E10:12_1_1_2"/><b>then </b>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>] implies ( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> ) )
 <b>by </b><i><a class="txt" href="topgen_5.html#E8:12_1_1_2"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E11:12_1_1_2"/>
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>] implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">a</font>) ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c5">a</font>] implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c5">a</font>) ) )
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:12_1_1_2"><i><font color="Green" title="E10">A8</font></i></a></i>;<br/><b>thus </b><a NAME="E12:12_1_1_2"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c5">a</font>] implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c5">a</font>) )
 <b>by </b><i><a class="txt" href="topgen_5.html#E8:12_1_1_2"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="topgen_5.html#E9:12_1_1_2"><i><font color="Green" title="E13">A12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
