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<span class="kw">let </span><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> (  <a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>) is  <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</a>  &amp;  <a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>) is  <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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"> (  <a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>) is  <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</a>  &amp;  <a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>) is  <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">S</font> =  <a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>);<br/>
<span class="kw">thus </span><a NAME="E1:69"/>
 <a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>) is  <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</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">  <a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>) is  <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">g</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="circcomb.html#D8" onmouseover="rs('circcomb/D8')" onmouseout="rh()">CIRCCOMB:def 8</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"> (  not <font color="Maroon" title="c5">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> or <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p3">(<span class="default"><a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1"/>
<font color="Maroon" title="c5">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>)</span>)</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"> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p3">(<span class="default"><a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<a NAME="E2:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circcomb.html#D5" onmouseover="rs('circcomb/D5')" onmouseout="rh()">CIRCCOMB:def 5</a></span>;<br/>
<a NAME="E3:69_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:69_1"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p3">(<span class="default"><a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="circcomb.html#D5" onmouseover="rs('circcomb/D5')" onmouseout="rh()">CIRCCOMB:def 5</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>

<span class="kw">thus </span><a NAME="E2:69"/>
 <a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>) is  <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">S9</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msafree2.html#D6" onmouseover="rs('msafree2/D6')" onmouseout="rh()">MSAFREE2:def 6</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"> (  not <font color="Maroon" title="c5">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font>) or  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">S9</font>  holds <br/>(  not  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_2"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c5">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K4" title="CIRCCOMB:func.4">1GateCircStr</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</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">  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">S9</font>  holds <br/>(  not  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">o1</font>, <font color="Maroon" title="c7">o2</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c5">S9</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"> (  not  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c6">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c7">o2</font> or <font color="Maroon" title="c6">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o2</font> )</span><br/>

<a NAME="E2:69_2"/>
<font color="Maroon" title="c6">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_2"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="circcomb.html#T37" onmouseover="rs('circcomb/T37')" onmouseout="rh()">CIRCCOMB:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:69_2"/>
(  not  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c6">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c7">o2</font> or <font color="Maroon" title="c6">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_2"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="circcomb.html#T37" onmouseover="rs('circcomb/T37')" onmouseout="rh()">CIRCCOMB:37</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>
