<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><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">f</font> be   <a href="funct_2.html#NM1">Function</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <a href="circcomb.html#K10">BOOLEAN</a> , <a href="circcomb.html#K10">BOOLEAN</a> ;<br/>







<b>assume </b><a NAME="E1:90"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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>
 ;<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"><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>,<font color="Maroon">c</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">S1</font> =  <a href="circcomb.html#K7">1GateCircStr</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>,<font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">S2</font> =  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6" target="_self">&lt;*</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>,<font color="Maroon">c</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>,<font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">R</font> = the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span>;<br/>
<a NAME="E2:90"/><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span> <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> <a href="xboole_0.html#K2">\/</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"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><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">c</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="circcomb.html#D2">CIRCCOMB:def 2</a></i>;<br/>
<i><font color="Green">E54</font></i>:  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K2">{</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 class="p2">(<span class="default"><a href="facirc_1.html#K13" target="_self">2GatesCircOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="txt" href="facirc_1.html#E4:90"><i><font color="Green">E55</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</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 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"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><span class="p5"><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">c</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#K2">}</a></span>

;<br/>
<a NAME="E4:90"/><i><font color="Green">E55</font></i>: 
 <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> <a href="hidden.html#R1">=</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>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>;<br/>
<a NAME="E5:90"/><b>then </b><i><font color="Green">E56</font></i>: 
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> <a href="hidden.html#R1">=</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> <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></i>;<br/>
<a NAME="E6:90"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="facirc_1.html#K6" target="_self">&lt;*</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>,<font color="Maroon">c</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</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>,<font color="Maroon">c</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="E7:90"/><b>then </b><i><font color="Green">E138</font></i>: 
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"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><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">c</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</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>,<font color="Maroon">c</font></span><a href="tarski.html#K2">}</a></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"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><span class="p5"><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">c</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></i>;<br/>
<b>thus </b><a NAME="E8:90"/>
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span><a href="enumset1.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:90_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E2:90_2"/><b>then </b><i><font color="Green">E140</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:90_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</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> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</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"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><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">c</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="facirc_1.html#T5" target="_self">Th5</a>, <a class="ref" href="facirc_1.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:90_2"/><b>then </b>
( <font color="Maroon">z</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> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</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> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</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>,<font color="Maroon">c</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</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"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><span class="p5"><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">c</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="facirc_1.html#T6" target="_self">Th6</a>, <a class="ref" href="facirc_1.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:90_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</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> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</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"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><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">c</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> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:90_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E9:90"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span><a href="enumset1.html#K1">}</a></span>
 ;<br/>

<a NAME="E10:90"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E11:90"/><b>then </b>
( ( <font color="Maroon">z</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; <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> <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"><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>,<font color="Maroon">c</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span> ) or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="facirc_1.html#T7" target="_self">Th7</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E12:90"/><b>then </b>
( (  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</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> &amp;  <a href="classes1.html#K6">the_rank_of</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> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</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"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><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">c</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> ) or ( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</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"><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>,<font color="Maroon">c</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span> ) )
 
<b>by </b><i>, <a class="ref" href="facirc_1.html#T7" target="_self">Th7</a>, <a class="ref" href="circcomb.html#T50">CIRCCOMB:50</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E13:90"/><b>then </b><i><font color="Green">E142</font></i>: 
(  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</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"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><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">c</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> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> )
 
<b>by </b><i><a class="txt" href="facirc_1.html#E6:90"><i><font color="Green">E57</font></i></a>, <a class="ref" href="circcomb.html#T50">CIRCCOMB:50</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E14:90"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><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">c</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>
 
;<br/>
<a NAME="E15:90"/><b>then </b><i><font color="Green">E143</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E16:90"/>
( <font color="Maroon">z</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> or <font color="Maroon">z</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"><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>,<font color="Maroon">c</font></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="facirc_1.html#T7" target="_self">Th7</a>, <a class="ref" href="facirc_1.html#T9" target="_self">Th9</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E17:90"/><b>then </b>
( <font color="Maroon">z</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> or <font color="Maroon">z</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"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><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">c</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="circcomb.html#T49">CIRCCOMB:49</a></i>;<br/>
<a NAME="E18:90"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E19:90"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div>
