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<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:13_1"/><i><font color="Green">E22</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E3:13_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:13_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:13_1_1"/>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">y</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">F<sub>1</sub></font>())
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:13_1_1"/><b>then </b>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">y</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:13_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:13_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>(), <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> )
 ;<br/><a NAME="E2:13_1_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>) &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:13_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  <a href="cohsp_1.html#V6">V169</a> <a href="funct_2.html#NM1">Function</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>(), <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="cohsp_1.html#D12">COHSP_1:def 12</a></i>;<br/>
<b>take </b><font color="Maroon">d</font> =  <a href="knaster.html#K1" target="_self">lfp</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">f</font>;<br/>

<a NAME="E7:13_1"/>
<font color="Maroon">d</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:13_1"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="abian.html#D3">ABIAN:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:13_1"/>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">d</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E10:13_1"/>
<font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


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