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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <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="c1">S</font><br/>  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b3">X</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Olive" title="b3">X</font></span>)</span>,<font color="Olive" title="b3">X</font><br/>  for <font color="Olive" title="b5">p</font> being   <a href="circcomb.html#NM2" title="CIRCCOMB:NM.2">FinSeqLen</a> of <font color="Olive" title="b2">n</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<font color="Olive" title="b5">p</font>,<font color="Olive" title="b4">f</font>) holds <br/><font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<font color="Olive" title="b5">p</font>,<font color="Olive" title="b4">f</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <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="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">X</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Olive" title="b2">X</font></span>)</span>,<font color="Olive" title="b2">X</font><br/>  for <font color="Olive" title="b4">p</font> being   <a href="circcomb.html#NM2" title="CIRCCOMB:NM.2">FinSeqLen</a> of <font color="Olive" title="b1">n</font>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<font color="Olive" title="b4">p</font>,<font color="Olive" title="b3">f</font>) holds <br/><font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<font color="Olive" title="b4">p</font>,<font color="Olive" title="b3">f</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" 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   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Olive" title="b1">X</font></span>)</span>,<font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b3">p</font> being   <a href="circcomb.html#NM2" title="CIRCCOMB:NM.2">FinSeqLen</a> of <font color="Maroon" title="c3">n</font>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<font color="Olive" title="b3">p</font>,<font color="Olive" title="b2">f</font>) holds <br/><font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<font color="Olive" title="b3">p</font>,<font color="Olive" title="b2">f</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">X</font></span>)</span>,<font color="Maroon" title="c4">X</font><br/>  for <font color="Olive" title="b2">p</font> being   <a href="circcomb.html#NM2" title="CIRCCOMB:NM.2">FinSeqLen</a> of <font color="Maroon" title="c3">n</font>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b1">f</font>) holds <br/><font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b1">f</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">X</font></span>)</span>,<font color="Maroon" title="c4">X</font>; <a class="txt" 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="circcomb.html#NM2" title="CIRCCOMB:NM.2">FinSeqLen</a> of <font color="Maroon" title="c3">n</font>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c5">f</font>) holds <br/><font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c5">f</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">p</font> be   <a href="circcomb.html#NM2" title="CIRCCOMB:NM.2">FinSeqLen</a> of <font color="Maroon" title="c3">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">f</font>) implies <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">f</font>) )</span><br/>











<span class="kw">assume </span><a NAME="E1:34"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">f</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">f</font>)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">X1</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c8">n1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c9">p1</font> being   <a href="circcomb.html#NM2" title="CIRCCOMB:NM.2">FinSeqLen</a> of <font color="Maroon" title="c8">n1</font>, <font color="Maroon" title="c10">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">n1</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c7">X1</font></span>)</span>,<font color="Maroon" title="c7">X1</font><span class="kw"> such that </span><br/><a NAME="E3:34"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c10">f1</font>)
 <span class="kw">and </span><br/><a NAME="E4:34"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c10">f1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circcmb3.html#D7" target="_self" title="CIRCCMB3:def.7">Def7</a></span>;<br/>
<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="c6">p</font>,<font color="Maroon" title="c5">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> = 
 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#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">f</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="circcomb.html#D6" title="CIRCCOMB:def.6">CIRCCOMB:def 6</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c10">f1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>
<br/>.= 
 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#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c10">f1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>
<br/>.= 
<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="c9">p1</font>,<font color="Maroon" title="c10">f1</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#D6" title="CIRCCOMB:def.6">CIRCCOMB:def 6</a></span>
;<br/>
<a NAME="E6:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c9">p1</font>,<font color="Maroon" title="c10">f1</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="zfmisc_1.html#T3" title="ZFMISC_1:th.3">ZFMISC_1:3</a></span>;<br/>
<a NAME="E7:34"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:34"/>
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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