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<div class="add">

<b>let </b><font color="Maroon">S1</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="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S2</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="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S</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="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>assume </b><a NAME="E1:50"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="circcomb.html#K3" target="_self">+*</a> <font color="Maroon">S2</font>
 ;<br/>

<b>let </b><font color="Maroon">A1</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">A2</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E2:50"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">S1</font> <a href="circcomb.html#R1" target="_self">tolerates</a> <font color="Maroon">S2</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font> )
 <b>and </b><br/><a NAME="E3:50"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="circcomb.html#K4" target="_self">+*</a> <font color="Maroon">A2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D3" target="_self">CIRCCOMB:def 3</a><br/>

<b>let </b><font color="Maroon">s</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">S</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">s1</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A1</font>;<br/><b>assume </b><a NAME="E1:50_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>
 ;<br/><div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font> ;<br/><a NAME="E3:50_1_1"/><i><font color="Green">E45</font></i>: 
(  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S</font> )
 <b>by </b><i>, <a class="txt" href="circcomb.html#E2:50"><i><font color="Green">E33</font></i></a>, </i>;<br/><a NAME="E4:50_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span> <a href="circuit1.html#K3">depends_on_in</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v1</font></span>)</span>,<font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v1</font></span>)</span> <a href="circuit1.html#K3">depends_on_in</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="circuit2.html#D5">CIRCUIT2:def 5</a></i>;<br/><a NAME="E5:50_1_1"/>
 <a href="msualg_1.html#K5">Den</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v1</font></span>)</span>,<font color="Maroon">A1</font> <a href="funct_2.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="circcomb.html#E2:50"><i><font color="Green">E33</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E6:50_1_1"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="txt" href="circcomb.html#E2:50"><i><font color="Green">E33</font></i></a>, <a class="ref" href="circcomb.html#T2" target="_self">Th2</a>, , <a class="txt" href="circcomb.html#E3:50"><i><font color="Green">E38</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">S1</font> ;<br/><a NAME="E3:50_1_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font>
 <b>by </b><i>, <a class="txt" href="circcomb.html#E2:50"><i><font color="Green">E33</font></i></a>, , </i>;<br/><a NAME="E4:50_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font> )
 <b>by </b><i>, <a class="ref" href="circuit2.html#D5">CIRCUIT2:def 5</a></i>;<br/><a NAME="E5:50_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/><b>hence </b><a NAME="E6:50_1_2"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="circcomb.html#T2" target="_self">Th2</a>, <a class="txt" href="circcomb.html#E3:50"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:50_1"/>
( ( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font> or ( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S</font> ) ) implies <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">s2</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A2</font>;<br/>

<b>assume </b><a NAME="E5:50"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 ;<br/>

<div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font> ;<br/><a NAME="E3:50_2"/><i><font color="Green">E120</font></i>: 
(  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:50_2"/><b>then </b><i><font color="Green">E133</font></i>: 
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span> <a href="circuit1.html#K3">depends_on_in</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v2</font></span>)</span>,<font color="Maroon">A2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v2</font></span>)</span> <a href="circuit1.html#K3">depends_on_in</a> <font color="Maroon">s2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="circuit2.html#D5">CIRCUIT2:def 5</a></i>;<br/><a NAME="E5:50_2"/>
 <a href="msualg_1.html#K5">Den</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v2</font></span>)</span>,<font color="Maroon">A2</font> <a href="funct_2.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="circcomb.html#E2:50"><i><font color="Green">E33</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E6:50_2"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="circcomb.html#T2" target="_self">Th2</a>, , <a class="txt" href="circcomb.html#E3:50"><i><font color="Green">E38</font></i></a>, <a class="txt" href="circcomb.html#E2"><i><font color="Green">Lemma29</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_3"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v</font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">S2</font> ;<br/><a NAME="E3:50_3"/>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:50_3"/><b>then </b><i><font color="Green">E135</font></i>: 
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v2</font> )
 <b>by </b><i>, <a class="ref" href="circuit2.html#D5">CIRCUIT2:def 5</a></i>;<br/><a NAME="E5:50_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/><b>hence </b><a NAME="E6:50_3"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="circcomb.html#T2" target="_self">Th2</a>, <a class="txt" href="circcomb.html#E3:50"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:50"/>
( ( not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font> &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S</font> ) ) or <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/>


</div>
