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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being   <a href="turing_1.html#L1" title="TURING_1:struct.1">TuringStr</a> <br/> for <font color="Olive" title="b2">s</font> being  <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Olive" title="b1">T</font>  st <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Olive" title="b1">T</font> holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">i</font></span><br/><b>let </b><font color="Maroon" title="c3">T</font> be    <a href="turing_1.html#L1" title="TURING_1:struct.1">TuringStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c3">T</font>  st <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c3">T</font> holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">i</font></span><br/><b>let </b><font color="Maroon" title="c4">s</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c3">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c3">T</font> implies <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">i</font> )</span><br/>







<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c3">T</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">i</font></span><br/>

<a NAME="E2:24"/><b>then </b>
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E3:24"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:24"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="turing_1.html#T14" target="_self" title="TURING_1:th.14">Th14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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