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<b>let </b><font color="Maroon">T</font> be    <a href="turing_1.html#L1" target="_self">TuringStr</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="turing_1.html#NM4">All-State</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">s</font> is <a href="turing_1.html#V2" target="_self">Accept-Halt</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E2:29"/><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:29"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:29"/><i><font color="Green">E34</font></i>: 
(  <a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">T</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E7:29"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>by </b><i/>;<br/>

<a NAME="E8:29"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:29"/>
 <a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , </i>;<br/>

<b>thus </b><a NAME="E10:29"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


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