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<b>let </b><font color="Maroon">s1</font> be   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1">SCM</a> , <font color="Maroon">s2</font> be   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1">SCM</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> , <font color="Maroon">c</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font> &amp;  <a href="scm_1.html#K2" target="_self">Complexity</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">s2</font> is <a href="ami_1.html#V9">halting</a> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:26"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="subset_1.html#M2">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="E5:26"/>
( <font color="Maroon">s2</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a>  &amp; <font color="Maroon">s2</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a>  )
 
<b>by </b><i><a class="ref" href="scm_1.html#T7" target="_self">Th7</a>, , </i>;<br/>
<a NAME="E6:26"/><b>then </b>
( <font color="Maroon">s2</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a>  &amp; <font color="Maroon">s2</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a>  )
 
<b>by </b><i><a class="ref" href="scm_1.html#T7" target="_self">Th7</a>, <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>;<br/>
<a NAME="E7:26"/><b>then </b>
( <font color="Maroon">s1</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a>  &amp; <font color="Maroon">s1</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a>  )
 
<b>by </b><i><a class="ref" href="scm_1.html#T7" target="_self">Th7</a>, <a class="ref" href="ami_1.html#T54">AMI_1:54</a></i>;<br/>
<b>hence </b> <a href="scm_1.html#K2" target="_self">Complexity</a> <font color="Maroon">s1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i/>
<br/>.= 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c</font>
<b>by </b><i/>
;<br/><br/>


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