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<div class="add">

<b>let </b><font color="Maroon" title="c1">N</font> be  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="ami_1.html#V2" title="AMI_1:attr.2">void</a> <a href="ami_1.html#V4" title="AMI_1:attr.4">halting</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a> <a href="ami_1.html#V9" title="AMI_1:attr.9">realistic</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c1">N</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">FinPartState</a> of <font color="Maroon" title="c2">S</font>;<br/>





<b>thus </b><a NAME="E1:84"/>
( <font color="Maroon" title="c3">p</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c3">p</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p1">(<span class="default"><a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <font color="Maroon" title="c3">p</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_1"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c3">p</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</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="ami_1.html#D29" target="_self" title="AMI_1:def.29">AMI_1:def 29</a><br/>

<b>assume </b><a NAME="E2:84_1"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c4">x</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> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:84_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E4:84_1"/><b>then </b><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="ami_1.html#E2:84_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:84_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="card_3.html#T66" title="CARD_3:th.66">CARD_3:66</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">x</font> as   <a href="ami_1.html#NM5" title="AMI_1:NM.5">FinPartState</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="ami_1.html#E4:84_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="ami_1.html#E81"><i><font color="Green" title="E24">Lm1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">s</font>
;<br/>

<b>thus </b><a NAME="E7:84_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font>
 ;<br/>

<b>thus </b><a NAME="E8:84_1"/>
<font color="Maroon" title="c3">p</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <font color="Maroon" title="c5">s</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E1:84_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="ami_1.html#E4:84_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></i>;<br/>

<a NAME="E9:84_1"/>
<span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="ami_1.html#E4:84_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>;<br/>
<b>hence </b><a NAME="E10:84_1"/>
<span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <font color="Maroon" title="c5">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="ami_1.html#E4:84_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></i>;<br/>


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

<a NAME="E2:84"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E3:84"/><b>then </b><i><font color="Green" title="E41">A4</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:84"/>
<font color="Maroon" title="c3">p</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p1">(<span class="default"><a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <font color="Maroon" title="c3">p</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">s</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">FinPartState</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E6:84"/><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E7:84"/><i><font color="Green" title="E43">A6</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <font color="Maroon" title="c4">s</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E8:84"/>
<span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K17" title="AMI_1:func.17">Result</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <font color="Maroon" title="c4">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="ami_1.html#E3:84"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="ami_1.html#D29" target="_self" title="AMI_1:def.29">Def29</a></i>;<br/>
<b>thus </b><a NAME="E9:84"/>
<font color="Maroon" title="c3">p</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E6:84"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="ami_1.html#E7:84"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></i>;<br/>


</div>
