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<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</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="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c5">C1</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">G1</font>;<br/><b>let </b><font color="Maroon" title="c6">C2</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="c2">G2</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:71"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="circtrm1.html#R4" title="CIRCTRM1:pred.4">form_embedding_of</a> <font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">C2</font>
 <b>and </b><br/><a NAME="E2:71"/><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">G2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="circtrm1.html#D11" target="_self" title="CIRCTRM1:def.11">CIRCTRM1:def 11</a><br/>

<b>let </b><font color="Maroon" title="c7">s2</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</font>;<br/><b>let </b><font color="Maroon" title="c8">s1</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font>;<br/>



<b>assume </b><a NAME="E3:71"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c8">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<a NAME="E4:71"/><i><font color="Green" title="E51">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font>
 
<b>by </b><i><a class="txt" href="circtrm1.html#E1:71"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="circtrm1.html#T42" target="_self" title="CIRCTRM1:th.42">Th42</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">v</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">G1</font>;<br/><b>assume </b><a NAME="E1:71_1"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">G1</font>
 ;<br/><a NAME="E2:71_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span>
 <b>by </b><i><a class="txt" href="circtrm1.html#E4:71"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E3:71_1"/>
<font color="Maroon" title="c7">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>
 <b>by </b><i><a class="txt" href="circtrm1.html#E2:71"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="facirc_1.html#T18" title="FACIRC_1:th.18">FACIRC_1:18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:71"/>
 <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c8">s1</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="c7">s2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="circtrm1.html#E1:71"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="circtrm1.html#E3:71"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="circtrm1.html#T46" target="_self" title="CIRCTRM1:th.46">Th46</a></i>;<br/>


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