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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K9">1GateCircuit</a> <span class="p2"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>,<font color="Maroon">f</font></span>)</span>;<br/>









<b>set </b><font color="Maroon">p</font> = <span class="p1"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>;<br/>
<a NAME="E1:81"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>,<font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E2:81"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="circcomb.html#D6">CIRCCOMB:def 6</a>, <a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>;<br/>
<a NAME="E3:81"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="circcomb.html#T64">CIRCCOMB:64</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>
<b>by </b><i><a class="ref" href="facirc_1.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_2.html#T145">FINSEQ_2:145</a></i>
;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="msafree2.html#NM1">Vertex</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>,<font color="Maroon">f</font></span>)</span> <b>by </b><i/>;<br/>
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>,<font color="Maroon">f</font></span>)</span> = 
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>
<b>by </b><i><a class="ref" href="circcomb.html#T49">CIRCCOMB:49</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>
;<br/>
<a NAME="E7:81"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>,<font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>,<font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:81"/>
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="circuit2.html#D5">CIRCUIT2:def 5</a></i>;<br/>


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