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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a> <a href="circcomb.html#V5">gate`2=den</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:66"/><i><font color="Green">E41</font></i>: 
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>1</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>2</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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D10">CIRCCOMB:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="circcomb.html#NM1">Gate</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E2:66"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="facirc_1.html#E17"><i><font color="Green">Lemma16</font></i></a></i>;<br/>
<a NAME="E3:66"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="circuit1.html#K3">depends_on_in</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="circuit1.html#D3">CIRCUIT1:def 3</a></i>;<br/>
<i><font color="Green">E44</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</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>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="facirc_1.html#E1:66"><i><font color="Green">E41</font></i></a>, <a class="txt" href="facirc_1.html#E2:66"><i><font color="Green">E42</font></i></a></i>
;<br/>
<a NAME="E5:66"/>
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <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="E6:66"/><b>then </b>
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="facirc_1.html#E2:66"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:66"/><b>then </b>
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E8:66"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/>
<a NAME="E9:66"/><b>then </b>
 <a href="msafree2.html#K5">action_at</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:66"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="facirc_1.html#E3:66"><i><font color="Green">E43</font></i></a>, <a class="txt" href="facirc_1.html#E4:66"><i><font color="Green">E44</font></i></a>, <a class="txt" href="facirc_1.html#E8:66"><i><font color="Green">E45</font></i></a>, <a class="ref" href="circuit2.html#D5">CIRCUIT2:def 5</a></i>;<br/>


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