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<b>let </b><font color="Maroon" title="c1">IIG</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a> <a href="msafree2.html#V5" title="MSAFREE2:attr.5">monotonic</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">SCS</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">IIG</font>;<br/><b>let </b><font color="Maroon" title="c3">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font>;<br/><b>let </b><font color="Maroon" title="c4">iv</font> be    <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</font>;<br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c4">iv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"> <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">s</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">IIG</font>
<b>by </b><i><a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c3">s</font></span>)</span>
<b>by </b><i><a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></i>
;<br/><b>hence </b><a NAME="E2:18_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">iv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c3">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:18"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E3:18_1"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">iv</font>
 ;<br/><a NAME="E4:18_1"/><i><font color="Green" title="E20">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">v</font> = <font color="Maroon" title="c5">x</font> as   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font> <b>by </b><i><a class="txt" href="circuit2.html#E3:18_1"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/><a NAME="E6:18_1"/>
<font color="Maroon" title="c4">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:18"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="circuit2.html#E3:18_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/><b>hence </b><a NAME="E7:18_1"/>
<font color="Maroon" title="c4">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="circuit2.html#E3:18_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="circuit2.html#E4:18_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="circuit2.html#D5" target="_self" title="CIRCUIT2:def.5">Def5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
<font color="Maroon" title="c4">iv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>


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