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<span class="kw">let </span><font color="Maroon" title="c4">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K21" title="TURING_1:func.21">UnionSt</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span>,<span class="p2">(<span class="default"> the <a href="turing_1.html#U1" title="TURING_1:sel.1">Symbols</a> of <font color="Maroon" title="c1">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="turing_1.html#U1" title="TURING_1:sel.1">Symbols</a> of <font color="Maroon" title="c2">t</font></span>)</span>,<span class="p2"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">p</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c1">s</font> ex <font color="Olive" title="b2">y</font> being   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">s</font> st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">s</font> ) &amp;  ex <font color="Olive" title="b1">q</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c2">t</font> ex <font color="Olive" title="b2">y</font> being   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c2">t</font> st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">s</font>,<font color="Olive" title="b1">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> implies ( <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K22" title="TURING_1:func.22">FirstTuringTran</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>,<span class="p1">(<span class="default"> the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K26" title="TURING_1:func.26">FirstTuringState</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K28" title="TURING_1:func.28">FirstTuringSymbol</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>) iff <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K23" title="TURING_1:func.23">SecondTuringTran</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>,<span class="p1">(<span class="default"> the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c2">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K27" title="TURING_1:func.27">SecondTuringState</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K29" title="TURING_1:func.29">SecondTuringSymbol</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:91_1_1"/>
(  ex <font color="Olive" title="b1">p</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c1">s</font> ex <font color="Olive" title="b2">y</font> being   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">s</font> st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">s</font> ) &amp;  ex <font color="Olive" title="b1">q</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c2">t</font> ex <font color="Olive" title="b2">y</font> being   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c2">t</font> st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">s</font>,<font color="Olive" title="b1">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> implies ( <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K22" title="TURING_1:func.22">FirstTuringTran</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>,<span class="p1">(<span class="default"> the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K26" title="TURING_1:func.26">FirstTuringState</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K28" title="TURING_1:func.28">FirstTuringSymbol</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>) iff <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K23" title="TURING_1:func.23">SecondTuringTran</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>,<span class="p1">(<span class="default"> the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c2">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K27" title="TURING_1:func.27">SecondTuringState</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K29" title="TURING_1:func.29">SecondTuringSymbol</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>) ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c5">p</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c1">s</font>, <font color="Maroon" title="c6">y</font> being   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">s</font><span class="kw"> such that </span><a NAME="E2:91_1_1_1"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">p</font>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:91_1_1_1"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  for <font color="Olive" title="b1">q</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c2">t</font><br/>  for <font color="Olive" title="b2">y</font> being   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c2">t</font>  holds  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">s</font>,<font color="Olive" title="b1">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or ( <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K22" title="TURING_1:func.22">FirstTuringTran</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>,<span class="p1">(<span class="default"> the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K26" title="TURING_1:func.26">FirstTuringState</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K28" title="TURING_1:func.28">FirstTuringSymbol</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>) iff <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K23" title="TURING_1:func.23">SecondTuringTran</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>,<span class="p1">(<span class="default"> the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c2">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K27" title="TURING_1:func.27">SecondTuringState</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K29" title="TURING_1:func.29">SecondTuringSymbol</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>) ) )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c7">q</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c2">t</font>, <font color="Maroon" title="c8">z</font> being   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c2">t</font><span class="kw"> such that </span><a NAME="E5:91_1_1_1"/><span class="lab"><font color="Green" title="E65">A3</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c7">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K22" title="TURING_1:func.22">FirstTuringTran</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>,<span class="p1">(<span class="default"> the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K26" title="TURING_1:func.26">FirstTuringState</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K28" title="TURING_1:func.28">FirstTuringSymbol</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>) iff <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K23" title="TURING_1:func.23">SecondTuringTran</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>,<span class="p1">(<span class="default"> the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c2">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K27" title="TURING_1:func.27">SecondTuringState</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K29" title="TURING_1:func.29">SecondTuringSymbol</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>) )</span><br/>

<a NAME="E6:91_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">p</font>, the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c7">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1_1_1"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E5:91_1_1_1"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:91_1_1_1"/>
( <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K22" title="TURING_1:func.22">FirstTuringTran</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>,<span class="p1">(<span class="default"> the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K26" title="TURING_1:func.26">FirstTuringState</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K28" title="TURING_1:func.28">FirstTuringSymbol</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>) iff <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K23" title="TURING_1:func.23">SecondTuringTran</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>,<span class="p1">(<span class="default"> the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c2">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K27" title="TURING_1:func.27">SecondTuringState</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="turing_1.html#K29" title="TURING_1:func.29">SecondTuringSymbol</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1_1_1"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
