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<span class="kw">let </span><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">s2</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</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="c1">T</font> ) &amp;  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</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="c1">T</font> ) implies <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c5">k1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><a NAME="E2:26_1_1"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
( <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</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="c1">T</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <br/>(  not <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font> or  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</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="c1">T</font> ) or <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c6">k2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><a NAME="E4:26_1_1"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
( <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">k2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">k2</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</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="c1">T</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font></span><br/>

<a NAME="E5:26_1_1"/>
( <font color="Maroon" title="c5">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k2</font> or <font color="Maroon" title="c6">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k1</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E6:26_1_1"/>
<font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="turing_1.html#T12" target="_self" onmouseover="rs('turing_1/T12')" onmouseout="rh()">Th12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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