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<div class="add">

<b>let </b><font color="Maroon">N</font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2" target="_self">void</a> <a href="ami_1.html#V4" target="_self">halting</a> <a href="ami_1.html#V5" target="_self">IC-Ins-separated</a> <a href="ami_1.html#V8" target="_self">definite</a> <a href="ami_1.html#V10" target="_self">realistic</a>  <a href="ami_1.html#L1" target="_self">AMI-Struct</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="ami_1.html#M1" target="_self">FinPartState</a> of <font color="Maroon">S</font>;<br/>





<b>thus </b><a NAME="E1:111"/>
( <font color="Maroon">p</font> is   <a href="ami_1.html#NM6">pre-program</a> of <font color="Maroon">S</font> implies <font color="Maroon">p</font> <a href="ami_1.html#R1" target="_self">computes</a> <a href="xboole_0.html#K1">{}</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18" target="_self">Result</a> <font color="Maroon">p</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:111_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">p</font> is   <a href="ami_1.html#NM6">pre-program</a> of <font color="Maroon">S</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" target="_self">AMI_1:def 29</a><br/>

<b>assume </b><a NAME="E2:111_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> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K18" target="_self">Result</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:111_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K18" target="_self">Result</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E4:111_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="ami_1.html#T15" target="_self">Th15</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">x</font> as    <a href="ami_1.html#M1" target="_self">FinPartState</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>thus </b><a NAME="E6:111_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<b>thus </b><a NAME="E7:111_1"/>
<font color="Maroon">p</font> <a href="ami_1.html#K17" target="_self">+*</a> <font color="Maroon">s</font> is   <a href="ami_1.html#NM6">pre-program</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="ami_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="funct_4.html#T22">FUNCT_4:22</a></i>;<br/>

<a NAME="E8:111_1"/>
<span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K18" target="_self">Result</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K18" target="_self">Result</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<b>hence </b><a NAME="E9:111_1"/>
<span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K18" target="_self">Result</a> <font color="Maroon">p</font></span>)</span></span>)</span> <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" target="_self">Result</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17" target="_self">+*</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T22">FUNCT_4:22</a></i>;<br/>


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

<a NAME="E2:111"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K18" target="_self">Result</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E3:111"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K18" target="_self">Result</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:111"/>
<font color="Maroon">p</font> <a href="ami_1.html#R1" target="_self">computes</a> <a href="xboole_0.html#K1">{}</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><a href="ami_1.html#K18" target="_self">Result</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being    <a href="ami_1.html#M1" target="_self">FinPartState</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:111"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E7:111"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">p</font> <a href="ami_1.html#K17" target="_self">+*</a> <font color="Maroon">s</font> is   <a href="ami_1.html#NM6">pre-program</a> of <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E8:111"/>
<span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K18" target="_self">Result</a> <font color="Maroon">p</font></span>)</span></span>)</span> <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" target="_self">Result</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ami_1.html#K17" target="_self">+*</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="ami_1.html#T14" target="_self">Th14</a>, </i>;<br/>
<b>thus </b><a NAME="E9:111"/>
<font color="Maroon">p</font> is   <a href="ami_1.html#NM6">pre-program</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="ami_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="funct_4.html#T22">FUNCT_4:22</a></i>;<br/>


</div>
