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<b>let </b><font color="Maroon">w</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K21" target="_self">UnionSt</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span>,<span class="p2">(<span class="default">the <a href="turing_1.html#U1" target="_self">Symbols</a> of <font color="Maroon">s</font> <a href="xboole_0.html#K2">\/</a> the <a href="turing_1.html#U1" target="_self">Symbols</a> of <font color="Maroon">t</font></span>)</span>,<span class="p2"><a href="domain_1.html#K8">{</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0,1</span><a href="domain_1.html#K8">}</a></span></span><a href="zfmisc_1.html#K3">:]</a></span>;<br/>

<b>thus </b><a NAME="E1:92_1_1"/>
(  ex <font color="Olive">p</font> being  <a href="turing_1.html#NM1">State</a> of <font color="Maroon">s</font> ex <font color="Olive">y</font> being  <a href="turing_1.html#NM3">Symbol</a> of <font color="Maroon">s</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">p</font>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">p</font> <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">s</font> ) &amp;  ex <font color="Olive">q</font> being  <a href="turing_1.html#NM1">State</a> of <font color="Maroon">t</font> ex <font color="Olive">y</font> being  <a href="turing_1.html#NM3">Symbol</a> of <font color="Maroon">t</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">s</font>,<font color="Olive">q</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> implies ( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="turing_1.html#K22" target="_self">FirstTuringTran</a> <font color="Maroon">s</font>,<font color="Maroon">t</font>,<span class="p1">(<span class="default">the <a href="turing_1.html#U3" target="_self">Tran</a> of <font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K26" target="_self">FirstTuringState</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K28" target="_self">FirstTuringSymbol</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span> iff <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="turing_1.html#K23" target="_self">SecondTuringTran</a> <font color="Maroon">s</font>,<font color="Maroon">t</font>,<span class="p1">(<span class="default">the <a href="turing_1.html#U3" target="_self">Tran</a> of <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K27" target="_self">SecondTuringState</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K29" target="_self">SecondTuringSymbol</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">p</font> being   <a href="turing_1.html#NM1">State</a> of <font color="Maroon">s</font>, <font color="Maroon">y</font> being   <a href="turing_1.html#NM3">Symbol</a> of <font color="Maroon">s</font><b> such that </b><a NAME="E1:92_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">p</font> <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">s</font> )
 ;<br/>

<b>given </b><font color="Maroon">q</font> being   <a href="turing_1.html#NM1">State</a> of <font color="Maroon">t</font>, <font color="Maroon">z</font> being   <a href="turing_1.html#NM3">Symbol</a> of <font color="Maroon">t</font><b> such that </b><a NAME="E2:92_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">s</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<a NAME="E3:92_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">s</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#D7" target="_self">Def7</a>, <a class="ref" href="turing_1.html#D8" target="_self">Def8</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E4:92_1_1_1"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="turing_1.html#K22" target="_self">FirstTuringTran</a> <font color="Maroon">s</font>,<font color="Maroon">t</font>,<span class="p1">(<span class="default">the <a href="turing_1.html#U3" target="_self">Tran</a> of <font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K26" target="_self">FirstTuringState</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K28" target="_self">FirstTuringSymbol</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span> iff <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="turing_1.html#K23" target="_self">SecondTuringTran</a> <font color="Maroon">s</font>,<font color="Maroon">t</font>,<span class="p1">(<span class="default">the <a href="turing_1.html#U3" target="_self">Tran</a> of <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K27" target="_self">SecondTuringState</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K29" target="_self">SecondTuringSymbol</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="turing_1.html#D7" target="_self">Def7</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>


</div>
