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<div class="add">

<b>let </b><font color="Maroon" title="c1">G2</font>, <font color="Maroon" title="c2">G1</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="c2">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="c1">G2</font>;<br/>











<b>assume </b><a NAME="E1:82"/><i><font color="Green" title="E55">A1</font></i>: 
<font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">C2</font> <a href="circtrm1.html#R5" title="CIRCTRM1:pred.5">are_similar_wrt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>
 ;<br/>

<a NAME="E2:82"/><b>then </b><i><font color="Green" title="E56">A2</font></i>: 
<font color="Maroon" title="c6">C2</font>,<font color="Maroon" title="c5">C1</font> <a href="circtrm1.html#R5" title="CIRCTRM1:pred.5">are_similar_wrt</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="ref" href="circtrm1.html#T40" target="_self" title="CIRCTRM1:th.40">Th40</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">s1</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font>;<br/><b>let </b><font color="Maroon" title="c8">s2</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</font>;<br/>



<b>assume </b><a NAME="E3:82"/><i><font color="Green" title="E57">A3</font></i>: 
<font color="Maroon" title="c7">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">v1</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">G1</font>;<br/>

<a NAME="E4:82"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon" title="c2">G1</font>,<font color="Maroon" title="c1">G2</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="circtrm1.html#E1:82"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="circtrm1.html#T38" target="_self" title="CIRCTRM1:th.38">Th38</a></i>;<br/>
<a NAME="E5:82"/><b>then </b><i><font color="Green" title="E59">A5</font></i>: 
( <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c2">G1</font>,<font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">G1</font> )
 
<b>by </b><i><a class="ref" href="circtrm1.html#D9" target="_self" title="CIRCTRM1:def.9">Def9</a></i>;<br/>
<a NAME="E6:82"/><b>then </b><i><font color="Green" title="E60">A6</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="c2">G1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <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">G2</font> )
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">v2</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font> as   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">G2</font> <b>by </b><i><a class="txt" href="circtrm1.html#E5:82"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="circtrm1.html#T31" target="_self" title="CIRCTRM1:th.31">Th31</a></i>;<br/>
<a NAME="E8:82"/>
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="circtrm1.html#E4:82"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="circtrm1.html#D9" target="_self" title="CIRCTRM1:def.9">Def9</a></i>;<br/>
<a NAME="E9:82"/><b>then </b><i><font color="Green" title="E61">A7</font></i>: 
<font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v2</font>
 
<b>by </b><i><a class="txt" href="circtrm1.html#E6:82"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>;<br/>
<b>thus </b><a NAME="E10:82"/>
( <font color="Maroon" title="c7">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c8">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">v1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_1"/><i><font color="Green" title="E62">A8</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c7">s1</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9" title="FACIRC_1:def.9">FACIRC_1:def 9</a><br/>

<b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9" title="FACIRC_1:def.9">FACIRC_1:def 9</a><br/>

<a NAME="E2:82_1"/>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c11">n</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#E1:82"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="circtrm1.html#E3:82"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="circtrm1.html#T56" target="_self" title="CIRCTRM1:th.56">Th56</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font>
<b>by </b><i><a class="txt" href="circtrm1.html#E6:82"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font>
<b>by </b><i><a class="txt" href="circtrm1.html#E1:82_1"><i><font color="Green" title="E62">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span>)</span>
<b>by </b><i><a class="txt" href="circtrm1.html#E3:82"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="circtrm1.html#E6:82"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><br/>


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

<b>assume </b><a NAME="E11:82"/><i><font color="Green" title="E62">A9</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c8">s2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9" title="FACIRC_1:def.9">FACIRC_1:def 9</a><br/>

<b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9" title="FACIRC_1:def.9">FACIRC_1:def 9</a><br/>

<a NAME="E12:82"/>
<font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="circtrm1.html#E1:82"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="circtrm1.html#E3:82"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="circtrm1.html#T52" target="_self" title="CIRCTRM1:th.52">Th52</a></i>;<br/>
<a NAME="E13:82"/><b>then </b>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="circtrm1.html#E2:82"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="circtrm1.html#T56" target="_self" title="CIRCTRM1:th.56">Th56</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font> = 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v2</font>
<b>by </b><i><a class="txt" href="circtrm1.html#E6:82"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="circtrm1.html#E9:82"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c8">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v2</font>
<b>by </b><i><a class="txt" href="circtrm1.html#E11:82"><i><font color="Green" title="E62">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font>
<b>by </b><i><a class="txt" href="circtrm1.html#E3:82"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="circtrm1.html#E6:82"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><br/>


</div>
