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<span class="kw">set </span><font color="Maroon" title="c1">S</font> =  <a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> (<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></font>());<br/>
<span class="kw">let </span><font color="Maroon" title="c2">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="circcmb3.html#K5" title="CIRCCMB3:func.5">1GateCircuit</a> (<span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></font>())</span>)</span>; <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">a1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><span class="p1">(<span class="default"><a href="circcmb3.html#K1" title="CIRCCMB3:func.1">Result</a> <font color="Maroon" title="c2">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#K3" title="CIRCCMB3:func.3">Output</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> (<span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></font>())</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">a1</font>)</span><br/>

 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">s</font> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> (<span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></font>())</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#T3" title="CIRCUIT1:th.3">CIRCUIT1:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></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/>.= 
<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> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></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="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a></span>
;<br/>
<a NAME="E2:74_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">a1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></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="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>1</sub></font>() implies <span class="p1">(<span class="default"><a href="circcmb3.html#K1" title="CIRCCMB3:func.1">Result</a> <font color="Maroon" title="c2">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#K3" title="CIRCCMB3:func.3">Output</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> (<span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></font>())</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a1</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E3:74_1"/>
<font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="circcmb3.html#K1" title="CIRCCMB3:func.1">Result</a> <font color="Maroon" title="c2">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#K3" title="CIRCCMB3:func.3">Output</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> (<span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></font>())</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a1</font>)</span><br/>

<a NAME="E4:74_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">a1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="finseq_2.html#T34" title="FINSEQ_2:th.34">FINSEQ_2:34</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="circcmb3.html#K1" title="CIRCCMB3:func.1">Result</a> <font color="Maroon" title="c2">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#K3" title="CIRCCMB3:func.3">Output</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> (<span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></font>())</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c2">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#K3" title="CIRCCMB3:func.3">Output</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> (<span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></font>())</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="circcmb3.html#T20" target="_self" title="CIRCCMB3:th.20">Th21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c2">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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon">F<sub>4</sub></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="circcmb3.html#T15" target="_self" title="CIRCCMB3:th.15">Th16</a></span>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon">F<sub>1</sub></font>()</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="circcomb.html#T56" title="CIRCCOMB:th.56">CIRCCOMB:56</a></span>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:74_1"><span class="lab"><font color="Green" title="E37">A3</font></span></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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