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<span class="kw">let </span><font color="Maroon" title="c1">s</font>, <font color="Maroon" title="c2">t</font> be    <a href="turing_1.html#L1" title="TURING_1:struct.1">TuringStr</a> ; <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">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="turing_1.html#K20" title="TURING_1:func.20">UnionSt</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">t</font>)  ex <font color="Olive" title="b2">x1</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="b3">x2</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c2">t</font> st <font color="Olive" title="b1">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"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">x2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="turing_1.html#K20" title="TURING_1:func.20">UnionSt</a> (<font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">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">x1</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">x2</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</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"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">q0</font> =  the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">t</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">p1</font> =  the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">s</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">A</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="turing_1.html#U2" title="TURING_1:sel.2">FStates</a> of <font color="Maroon" title="c1">s</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> 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#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">B</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</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></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, the <a href="turing_1.html#U2" title="TURING_1:sel.2">FStates</a> of <font color="Maroon" title="c2">t</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_1"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="turing_1.html#U2" title="TURING_1:sel.2">FStates</a> of <font color="Maroon" title="c1">s</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> 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#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</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></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, the <a href="turing_1.html#U2" title="TURING_1:sel.2">FStates</a> of <font color="Maroon" title="c2">t</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><span class="kw">suppose </span></a><a NAME="E1:82_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="turing_1.html#U2" title="TURING_1:sel.2">FStates</a> of <font color="Maroon" title="c1">s</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> 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#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</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">  ex <font color="Olive" title="b1">x1</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">x2</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</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"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x1</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="c9">x2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> 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#K6" title="DOMAIN_1:func.6">}</a></span><span class="kw"> such that </span><br/><a NAME="E3:82_1_1"/><span class="lab"><font color="Green" title="E57">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"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</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="ref" href="domain_1.html#T1" onmouseover="rs('domain_1/T1')" onmouseout="rh()">DOMAIN_1:1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c8">x1</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">x2</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</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"><font color="Maroon" title="c8">x1</font>,<font color="Olive" title="b1">x2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">take </span>
 the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">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">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"><font color="Maroon" title="c8">x1</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></span><br/><span class="kw">thus </span><a NAME="E4:82_1_1"/>
<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"><font color="Maroon" title="c8">x1</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1_1"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><span class="kw">suppose </span></a><a NAME="E1:82_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</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></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, the <a href="turing_1.html#U2" title="TURING_1:sel.2">FStates</a> of <font color="Maroon" title="c2">t</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</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">  ex <font color="Olive" title="b1">x1</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">x2</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</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"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</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></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c9">x2</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c2">t</font><span class="kw"> such that </span><br/><a NAME="E3:82_1_2"/><span class="lab"><font color="Green" title="E57">A2</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"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</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="ref" href="domain_1.html#T1" onmouseover="rs('domain_1/T1')" onmouseout="rh()">DOMAIN_1:1</a></span>;<br/><span class="kw">take </span>
 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" 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">x2</font> being   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</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"> 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">x2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">take </span>
<font color="Maroon" title="c9">x2</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">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"> 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="c9">x2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">thus </span><a NAME="E4:82_1_2"/>
<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"> 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="c9">x2</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="#E3:82_1_2"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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