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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon">F<sub>2</sub></font>()) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() = <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default">$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>);<br/>
<b>consider </b><font color="Maroon" title="c1">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E2:41_1"/><i><font color="Green" title="E22">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Maroon" title="c1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c2">S</font> =  <a href="circcomb.html#K6" title="CIRCCOMB:func.6">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>,<font color="Maroon" title="c1">g</font> as  <a href="circcmb3.html#V4" title="CIRCCMB3:attr.4">one-gate</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c2">S</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c3">A</font> =  <a href="circcomb.html#K8" title="CIRCCOMB:func.8">1GateCircuit</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>,<font color="Maroon" title="c1">g</font> as  <a href="circcmb3.html#V5" title="CIRCCMB3:attr.5">one-gate</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">A</font>
;<br/>

<a NAME="E5:41_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/>
<b>hence </b><a NAME="E6:41_1"/>
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="circcomb.html#T49" title="CIRCCOMB:th.49">CIRCCOMB:49</a></i>;<br/>

<b>let </b><font color="Maroon" title="c4">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c3">A</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">sx</font> = <font color="Maroon" title="c4">s</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="ref" href="circcmb3.html#T13" target="_self" title="CIRCCMB3:th.13">Th13</a></i>;<br/>
<a NAME="E8:41_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E9:41_1"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="circcmb3.html#K1" title="CIRCCMB3:func.1">Result</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K5" title="CIRCCMB3:func.5">Output</a> <font color="Maroon" title="c2">S</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K5" title="CIRCCMB3:func.5">Output</a> <font color="Maroon" title="c2">S</font></span>)</span>
<b>by </b><i><a class="ref" href="circcmb3.html#T21" target="_self" title="CIRCCMB3:th.21">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>,<font color="Maroon" title="c1">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="circcmb3.html#T16" target="_self" title="CIRCCMB3:th.16">Th16</a></i>
<br/>.= 
<font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="circcomb.html#T64" title="CIRCCOMB:th.64">CIRCCOMB:64</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">sx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>)
<b>by </b><i><a class="txt" href="circcmb3.html#E2:41_1"><i><font color="Green" title="E22">A1</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="circcmb3.html#E9:41_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>)
<b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>
;<br/>


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