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<b>let </b><font color="Maroon">G2</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>let </b><font color="Maroon">G1</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>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">C1</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">G1</font>;<br/><b>let </b><font color="Maroon">C2</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">G2</font>;<br/>











<b>assume </b><a NAME="E1:72"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="circtrm1.html#R4" target="_self">form_embedding_of</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font>
 ;<br/>

<a NAME="E2:72"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">G1</font>,<font color="Maroon">G2</font>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">s2</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">C2</font>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">C1</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E3:72"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:72"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">v</font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">G1</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">G1</font> holds <br/><font color="Maroon">s2</font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">s2'</font> = <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#NK3" target="_self">*</a> <font color="Maroon">f</font> as   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">C1</font> <b>by </b><i><a class="ref" href="circtrm1.html#T5" target="_self">Th5</a>, <a class="ref" href="circtrm1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:72"/><i><font color="Green">E71</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> )
 
<b>by </b><i>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">G1</font>;<br/><a NAME="E1:72_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">s2'</font> <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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="circtrm1.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">fv</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">G2</font> <b>by </b><i>, </i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:72_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">G1</font>
 ;<br/><a NAME="E2:72_1_1"/><b>then </b>
( <font color="Maroon">s2</font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp;  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,1 <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K6">Following</a> <font color="Maroon">s2</font> )
 <b>by </b><i>, <a class="ref" href="facirc_1.html#T14">FACIRC_1:14</a></i>;<br/><b>hence </b><font color="Maroon">s2'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> = 
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, <a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a></i>
<br/>.= 
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="circtrm1.html#T6" target="_self">Th6</a>, <a class="ref" href="circtrm1.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><br/></div>
<b>end;</b></div><a NAME="E4:72_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">G1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">G2</font>
 <b>by </b><i>, </i>;<br/><b>assume </b><a NAME="E5:72_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">G1</font>
 ;<br/><a NAME="E6:72_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">fv</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">G1</font></span>)</span>
 <b>by </b><i><a class="ref" href="circtrm1.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:72_1"/><i><font color="Green">E105</font></i>: 
 <a href="msafree2.html#K5">action_at</a> <font color="Maroon">fv</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="circtrm1.html#E1:72"><i><font color="Green">E36</font></i></a></i>;<br/><b>thus </b><font color="Maroon">s2'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> = 
<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">fv</font></span>)</span>,<font color="Maroon">C2</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">fv</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/>.= 
<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">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">fv</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="circtrm1.html#T5" target="_self">Th5</a>, , </i>
<br/>.= 
<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">C1</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">s1</font></span>)</span>
<b>by </b><i><a class="ref" href="circtrm1.html#T5" target="_self">Th5</a>, <a class="ref" href="circtrm1.html#T6" target="_self">Th6</a>, , <a class="ref" href="circtrm1.html#T3" target="_self">Th3</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:72"/>
 <a href="circuit2.html#K6">Following</a> <font color="Maroon">s1</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#NK3" target="_self">*</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="circuit2.html#D5">CIRCUIT2:def 5</a></i>;<br/>


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