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<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#V7" target="_self">steady-programmed</a> <a href="ami_1.html#V8" target="_self">definite</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">s</font> be   <a href="ami_1.html#NM4">State</a> of <font color="Maroon">S</font>;<br/>





<b>given </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:79"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">s</font> <a href="ami_1.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" target="_self">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5" target="_self">halt</a> <font color="Maroon">S</font>
 ;<br/>

<b>set </b><font color="Maroon">s'</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">k</font>;<br/>
<a NAME="E2:79"/><i><font color="Green">E34</font></i>: 
 <a href="ami_1.html#K8" target="_self">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5" target="_self">halt</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="ami_1.html#D14" target="_self">Def14</a>, </i>;<br/>
<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1"><b>suppose </b></a><a NAME="E1:79_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:79_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:79_1_1_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="ami_1.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E6:79_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">i</font> is <a href="ami_1.html#V9" target="_self">halting</a>
 <b>by </b><i><a class="ref" href="ami_1.html#T7" target="_self">Th7</a>, </i>;<br/><b>thus </b> <a href="ami_1.html#K12" target="_self">Result</a> <font color="Maroon">s</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="ami_1.html#D14" target="_self">Def14</a>, </i>
<br/>.= 
 <a href="ami_1.html#K12" target="_self">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T7" target="_self">Th7</a>, , <a class="ref" href="ami_1.html#T9" target="_self">Th9</a>, <a class="ref" href="ami_1.html#D13" target="_self">Def13</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2"><b>suppose </b></a><a NAME="E1:79_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="ami_1.html#T7" target="_self">Th7</a>, <a class="ref" href="ami_1.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E3:79_1_1_2"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E4:79_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">i</font> is <a href="ami_1.html#V9" target="_self">halting</a>
 <b>by </b><i><a class="ref" href="ami_1.html#T7" target="_self">Th7</a>, <a class="ref" href="ami_1.html#T8" target="_self">Th8</a>, </i>;<br/><b>thus </b> <a href="ami_1.html#K12" target="_self">Result</a> <font color="Maroon">s</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="ami_1.html#D14" target="_self">Def14</a>, </i>
<br/>.= 
 <a href="ami_1.html#K12" target="_self">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T7" target="_self">Th7</a>, <a class="ref" href="ami_1.html#T8" target="_self">Th8</a>, , <a class="ref" href="ami_1.html#T9" target="_self">Th9</a>, <a class="ref" href="ami_1.html#D13" target="_self">Def13</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:79"/>
 <a href="ami_1.html#K12" target="_self">Result</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K12" target="_self">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 ;<br/>


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