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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="E1:51"/><i><font color="Green">E24</font></i>: 
(  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font> &amp; <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/>

<a NAME="E2:51"/><b>then </b><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 
<b>by </b><i/>;<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="E3:51"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A1</font> <a href="circcomb.html#R2" target="_self">tolerates</a> <font color="Maroon">A2</font>
 <b>and </b><br/><a NAME="E4:51"/><i><font color="Green">E42</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>
 ;<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">s1</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A1</font>;<br/><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="E5:51"/><i><font color="Green">E43</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> &amp; <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/>

<a NAME="E6:51"/><i><font color="Green">E45</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s2</font></span>)</span> <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s2</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">x</font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="circcomb.html#E2:51"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circcomb.html#E3:51"><i><font color="Green">E38</font></i></a>, </i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:51_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s2</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">x</font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">S2</font> <b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/><a NAME="E3:51_1_1"/>
<span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font></span>)</span> <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="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E4:51_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font> or <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><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_1_1"/>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font>
 ;<br/><a NAME="E2:51_1_1_1"/><b>then </b>
( <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font></span>)</span> &amp; not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font> )
 <b>by </b><i><a class="txt" href="circcomb.html#E2:51"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E3:51_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
( not <font color="Maroon">v2</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">S2</font> &amp; not <font color="Maroon">v2</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">S1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:51_1_1_1"/>
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_4.html#K1">+*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font>
 <b>by </b><i><a class="txt" href="circcomb.html#E2:51"><i><font color="Green">E33</font></i></a>, </i>;<br/><a NAME="E5:51_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/><a NAME="E6:51_1_1_1"/><b>then </b>
not <font color="Maroon">v</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">S</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:51_1_1_1"/><b>then </b>
<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">S</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:51_1_1_1"/>
<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/></div><b>end;</b></div><a NAME="E6:51_1_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</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">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> ) )
 <b>by </b><i><a class="ref" href="circcomb.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E7:51_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">x</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">x</font>
 <b>by </b><i><a class="txt" href="circcomb.html#E2:51"><i><font color="Green">E33</font></i></a>, <a class="ref" href="circcomb.html#T2" target="_self">Th2</a>, , , </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E4:51_1"/><i><font color="Green">E107</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s2</font></span>)</span>
 ;<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> <b>by </b><i>, <a class="txt" href="circcomb.html#E3:51"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:51_1"/>
<span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font></span>)</span> <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="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E7:51_1"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font> or <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="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:51_1"/>
 <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E9:51_1"/><b>then </b><i><font color="Green">E109</font></i>: 
not <font color="Maroon">v1</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">S2</font>
 <b>by </b><i><a class="txt" href="circcomb.html#E3:51"><i><font color="Green">E38</font></i></a>, <a class="ref" href="circcomb.html#T3" target="_self">Th3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_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>
 ;<br/><a NAME="E2:51_1_2"/><b>then </b>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font></span>)</span>
 ;<br/><a NAME="E3:51_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
not <font color="Maroon">v1</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">S1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:51_1_2"/>
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_4.html#K1">+*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font>
 <b>by </b><i><a class="txt" href="circcomb.html#E2:51"><i><font color="Green">E33</font></i></a>, </i>;<br/><a NAME="E5:51_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/><a NAME="E6:51_1_2"/><b>then </b>
not <font color="Maroon">v</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">S</font>
 <b>by </b><i>, <a class="txt" href="circcomb.html#E4:51"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:51_1_2"/><b>then </b>
<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">S</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:51_1_2"/>
<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/></div><b>end;</b></div><b>hence </b><a NAME="E11:51_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">x</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">x</font>
 <b>by </b><i><a class="txt" href="circcomb.html#E2:51"><i><font color="Green">E33</font></i></a>, <a class="ref" href="circcomb.html#T2" target="_self">Th2</a>, , , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:51"/>
 <a href="circuit2.html#K6">Following</a> <font color="Maroon">s</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_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s2</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="circcomb.html#E3:51"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>


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