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<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:57_1_1"/><i><font color="Green">E28</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V3">constant</a> &amp;  <a href="yellow_6.html#K1">the_value_of</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="circcomb.html#V4">gate`2=den</a> )
 <b>by </b><i><a class="ref" href="circcmb3.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E3:57_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #);<br/>
<a NAME="E4:57_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> is <a href="pre_circ.html#V1">locally-finite</a>
 
<b>by </b><i><a class="ref" href="msafree2.html#D11">MSAFREE2:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #) as   <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msafree2.html#D11">MSAFREE2:def 11</a></i>;<br/>
<a NAME="E6:57_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E2:57_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="circcomb.html#D10">CIRCCOMB:def 10</a></i>;<br/>
<a NAME="E7:57_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> is <a href="circcomb.html#V4">gate`2=den</a> &amp; not the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V3">constant</a> &amp;  <a href="yellow_6.html#K1">the_value_of</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="circcmb3.html#E2:57_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="circcomb.html#D10">CIRCCOMB:def 10</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="circcmb3.html#M2" target="_self">Circuit</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="circcmb3.html#D10" target="_self">Def10</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>9</sub></font> being    <a href="circcomb.html#M1">FinSeqLen</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>8</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E10:57_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="circcmb3.html#K6" target="_self">1GateCircStr</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="circcmb3.html#D6" target="_self">Def6</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="circcomb.html#K9">1GateCircuit</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E11:57_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <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"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:57_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E10:57_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="circcomb.html#D6">CIRCCOMB:def 6</a></i>;<br/>
<a NAME="E13:57_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="circcomb.html#D10">CIRCCOMB:def 10</a></i>;<br/>
<a NAME="E14:57_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <span class="p3">(<span class="default"><a href="circcomb.html#K9">1GateCircuit</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E10:57_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="circcmb3.html#E12:57_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="circcomb.html#D10">CIRCCOMB:def 10</a></i>;<br/>
<a NAME="E15:57_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E13:57_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E16:57_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:57_1_1"/>
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:57_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E12:57_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:57_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="circcmb3.html#E2:57_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="yellow_6.html#D1">YELLOW_6:def 1</a></i>;<br/>
<a NAME="E21:57_1_1"/><i><font color="Green">E37</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R6">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E2:57_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="circcmb3.html#E3:57_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="circcmb3.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E22:57_1_1"/>
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E12:57_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E23:57_1_1"/><b>then </b>
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E24:57_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E2:57_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="circcmb3.html#E20:57_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>;<br/>
<a NAME="E25:57_1_1"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E12:57_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E26:57_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E18:57_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="circcmb3.html#E24:57_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:57_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E28:57_1_1"/>
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="txt" href="circcmb3.html#E12:57_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<i><font color="Green">E41</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="circcmb3.html#E12:57_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="circcmb3.html#E26:57_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="circcmb3.html#E27:57_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="circcmb3.html#E21:57_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="circcomb.html#T6">CIRCCOMB:6</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_2_1"><b>suppose </b></a><a NAME="E1:57_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:57_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="circcmb3.html#E16:57_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="circcmb3.html#E30:57_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="circcmb3.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E3:57_1_1_2_1"/><b>then </b><i><font color="Green">E42</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K9">1GateCircuit</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circcmb3.html#E10:57_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="circcmb3.html#E21:57_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="circcomb.html#D14">CIRCCOMB:def 14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:57_1_1_2_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:57_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></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"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="circcmb3.html#E10:57_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="circcomb.html#D6">CIRCCOMB:def 6</a></i>;<br/><a NAME="E3:57_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:57_1_1_2_1_1"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K9">1GateCircuit</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="circcmb3.html#E13:57_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="circcmb3.html#E14:57_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:57_1_1_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> is <a href="msualg_1.html#V4">strict</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="circcmb3.html#V5" target="_self">one-gate</a> )
 <b>by </b><i><a class="txt" href="circcmb3.html#E10:57_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="circcmb3.html#E3:57_1_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_2_2"><b>suppose </b></a><a NAME="E1:57_1_1_2_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:57_1_1_2_2"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="circcmb3.html#E12:57_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E3:57_1_1_2_2"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><font color="Maroon">c<sub>8</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="circcmb3.html#E1:57_1_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="circcmb3.html#E1:57_1_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>
;<br/><a NAME="E5:57_1_1_2_2"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:57_1_1_2_2"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="circcmb3.html#E13:57_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>8</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="circcmb3.html#E18:57_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="circcmb3.html#E24:57_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="circcmb3.html#E3:57_1_1_2_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="circcmb3.html#E5:57_1_1_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="circcomb.html#K9">1GateCircuit</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>15</sub></font>;<br/><a NAME="E8:57_1_1_2_2"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <span class="p3">(<span class="default"><a href="circcomb.html#K9">1GateCircuit</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="circcmb3.html#E10:57_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="circcmb3.html#E12:57_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="circcomb.html#D10">CIRCCOMB:def 10</a></i>;<br/><a NAME="E9:57_1_1_2_2"/><i><font color="Green">E46</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K9">1GateCircuit</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circcmb3.html#E10:57_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="circcmb3.html#E21:57_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="circcomb.html#D14">CIRCCOMB:def 14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_2_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:57_1_1_2_2_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:57_1_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></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"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="circcmb3.html#E10:57_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="circcomb.html#D6">CIRCCOMB:def 6</a></i>;<br/><a NAME="E3:57_1_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:57_1_1_2_2_2"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K9">1GateCircuit</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="circcmb3.html#E13:57_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="circcmb3.html#E8:57_1_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:57_1_1_2_2"/>
( <font color="Maroon">c<sub>6</sub></font> is <a href="msualg_1.html#V4">strict</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="circcmb3.html#V5" target="_self">one-gate</a> )
 <b>by </b><i><a class="txt" href="circcmb3.html#E10:57_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="circcmb3.html#E9:57_1_1_2_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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