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

<b>let </b><font color="Maroon">F</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> , <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green">E31</font></i>: 
 <a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F</font> is <a href="ami_5.html#V2">data-only</a>
 ;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> <a href="ami_1.html#R1">computes</a> <font color="Maroon">F</font>
 ;<br/><b>thus </b><a NAME="E2:20_1"/>
 <a href="scmfsa_5.html#K1" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font> <a href="ami_1.html#R1">computes</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_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="ami_1.html#D29">AMI_1:def 29</a><br/>

<b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:20_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">x</font> as  <a href="ami_5.html#V1">data-only</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a>  <b>by </b><i>, , <a class="ref" href="ami_3.html#T32">AMI_3:32</a>, <a class="ref" href="ami_5.html#D9">AMI_5:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>thus </b><a NAME="E4:20_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<b>consider </b><font color="Maroon">s1</font> being    <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> <b> such that </b><br/><a NAME="E6:20_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s1</font> is   <a href="ami_1.html#NM6">pre-program</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a>  &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K18">Result</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="ami_1.html#D29">AMI_1:def 29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Fs1</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> as    <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a>  <b>by </b><i>, <a class="ref" href="ami_5.html#T61">AMI_5:61</a></i>;<br/>
<a NAME="E8:20_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">Fs1</font> is <a href="ami_5.html#V1">data-only</a>
 
<b>by </b><i>, , , <a class="ref" href="ami_5.html#D9">AMI_5:def 9</a></i>;<br/>
<a NAME="E9:20_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="ami_5.html#K7">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18">Result</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="ami_5.html#T74">AMI_5:74</a></i>;<br/>
<a NAME="E10:20_1_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa_5.html#K1" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_5.html#K1" target="_self">Relocated</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:20_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E12:20_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E13:20_1_1"/>
<span class="p1">(<span class="default"><a href="scmfsa_5.html#K1" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font> is   <a href="ami_1.html#NM6">pre-program</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i>, , , </i>;<br/>

 <a href="ami_5.html#K7">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18">Result</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s1</font></span>)</span></span>)</span> = 
 <a href="ami_5.html#K7">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18">Result</a> <span class="p2">(<span class="default"><a href="scmfsa_5.html#K1" target="_self">Relocated</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="scmfsa_5.html#E2"><i><font color="Green">Lemma27</font></i></a>, </i>
<br/>.= 
 <a href="ami_5.html#K7">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18">Result</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_5.html#K1" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E15:20_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K18">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_5.html#K1" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="ami_5.html#T74">AMI_5:74</a></i>;<br/>


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

<b>assume </b><a NAME="E4:20"/><i><font color="Green">E55</font></i>: 
 <a href="scmfsa_5.html#K1" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font> <a href="ami_1.html#R1">computes</a> <font color="Maroon">F</font>
 ;<br/>

<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="ami_1.html#D29">AMI_1:def 29</a><br/>

<b>assume </b><a NAME="E5:20"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E6:20"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">x</font> as  <a href="ami_5.html#V1">data-only</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a>  <b>by </b><i>, , <a class="ref" href="ami_3.html#T32">AMI_3:32</a>, <a class="ref" href="ami_5.html#D9">AMI_5:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>thus </b><a NAME="E8:20"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<b>consider </b><font color="Maroon">s1</font> being    <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> <b> such that </b><br/><a NAME="E10:20"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <span class="p1">(<span class="default"><a href="scmfsa_5.html#K1" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s1</font> is   <a href="ami_1.html#NM6">pre-program</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a>  &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K18">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_5.html#K1" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="ami_1.html#D29">AMI_1:def 29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Fs1</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> as    <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a>  <b>by </b><i>, <a class="ref" href="ami_5.html#T61">AMI_5:61</a></i>;<br/>
<a NAME="E12:20"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">Fs1</font> is <a href="ami_5.html#V1">data-only</a>
 
<b>by </b><i>, , , <a class="ref" href="ami_5.html#D9">AMI_5:def 9</a></i>;<br/>
<a NAME="E13:20"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="ami_5.html#K7">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18">Result</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_5.html#K1" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="ami_5.html#T74">AMI_5:74</a></i>;<br/>
<a NAME="E14:20"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa_5.html#K1" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_5.html#K1" target="_self">Relocated</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:20"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E16:20"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E17:20"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font> is <a href="ami_1.html#V11">autonomic</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E18:20"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font> is <a href="ami_1.html#V12">halting</a>
 
<b>by </b><i>, , <a class="txt" href="scmfsa_5.html#E2"><i><font color="Green">Lemma27</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E19:20"/>
<font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font> is   <a href="ami_1.html#NM6">pre-program</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i>, , <a class="txt" href="scmfsa_5.html#E2"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="scmfsa_5.html#T1" target="_self">Th1</a>, </i>;<br/>

 <a href="ami_5.html#K7">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18">Result</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_5.html#K1" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s1</font></span>)</span></span>)</span> = 
 <a href="ami_5.html#K7">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18">Result</a> <span class="p2">(<span class="default"><a href="scmfsa_5.html#K1" target="_self">Relocated</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="ami_5.html#K7">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18">Result</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_5.html#E2"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="scmfsa_5.html#T1" target="_self">Th1</a>, , </i>
;<br/>
<b>hence </b><a NAME="E21:20"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K18">Result</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17">+*</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="ami_5.html#T74">AMI_5:74</a></i>;<br/>


</div>
