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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="E1:84"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">C1</font>,<font color="Maroon">C2</font> <a href="circtrm1.html#R5" target="_self">are_similar_wrt</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:84"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">C2</font>,<font color="Maroon">C1</font> <a href="circtrm1.html#R5" target="_self">are_similar_wrt</a> <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> ,<font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i/>;<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>let </b><font color="Maroon">s2</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">C2</font>;<br/>



<b>assume </b><a NAME="E3:84"/><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>
 ;<br/>

<b>let </b><font color="Maroon">v1</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">G1</font>;<br/>

<a NAME="E4:84"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">G1</font>,<font color="Maroon">G2</font> <a href="circtrm1.html#R1" target="_self">are_equivalent_wrt</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 
<b>by </b><i>, <a class="txt" href="circtrm1.html#E3:84"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E5:84"/><b>then </b><i><font color="Green">E71</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> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> ,<font color="Maroon">g</font> <a href="funct_1.html#K2">"</a>  <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">G2</font>,<font color="Maroon">G1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:84"/><b>then </b><i><font color="Green">E72</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#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</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/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">G2</font> <b>by </b><i>, </i>;<br/>
<a NAME="E8:84"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:84"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>
<b>thus </b><a NAME="E10:84"/>
( <font color="Maroon">s1</font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">v1</font> implies <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">v1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<font color="Olive">n</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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a><br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a><br/>

<a NAME="E2:84_1"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">n</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">v1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
<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">v1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E11:84"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<font color="Olive">n</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">v1</font></span>)</span> <a href="hidden.html#R1">=</a> <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">v1</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a><br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a><br/>

<a NAME="E12:84"/>
<font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:84"/><b>then </b>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font> = 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v2</font>
<b>by </b><i>, <a class="ref" href="circtrm1.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v2</font>
<b>by </b><i/>
<br/>.= 
<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="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><br/>


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