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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="struct_0.html#V11" title="STRUCT_0:attr.11">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b3">C1</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b4">C2</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b3">C1</font>,<font color="Olive" title="b4">C2</font> <a href="circtrm1.html#R5" title="CIRCTRM1:pred.5">are_similar_wrt</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> holds <br/>for <font color="Olive" title="b5">s1</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b3">C1</font><br/> for <font color="Olive" title="b6">s2</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b4">C2</font>  st <font color="Olive" title="b5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font> holds <br/>( <font color="Olive" title="b5">s1</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> iff <font color="Olive" title="b6">s2</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C1</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">C2</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">C1</font>,<font color="Olive" title="b2">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> holds <br/>for <font color="Olive" title="b3">s1</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b1">C1</font><br/> for <font color="Olive" title="b4">s2</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b2">C2</font>  st <font color="Olive" title="b3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/>( <font color="Olive" title="b3">s1</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> iff <font color="Olive" title="b4">s2</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> )</span><br/><b>let </b><font color="Maroon" title="c5">C1</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C2</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c5">C1</font>,<font color="Olive" title="b1">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> holds <br/>for <font color="Olive" title="b2">s1</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font><br/> for <font color="Olive" title="b3">s2</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b1">C2</font>  st <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/>( <font color="Olive" title="b2">s1</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> iff <font color="Olive" title="b3">s2</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> )</span><br/><b>let </b><font color="Maroon" title="c6">C2</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies for <font color="Olive" title="b1">s1</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font><br/> for <font color="Olive" title="b2">s2</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</font>  st <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/>( <font color="Olive" title="b1">s1</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> iff <font color="Olive" title="b2">s2</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> ) )</span><br/>











<b>assume </b><a NAME="E1:80"/><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s1</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font><br/> for <font color="Olive" title="b2">s2</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</font>  st <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/>( <font color="Olive" title="b1">s1</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> iff <font color="Olive" title="b2">s2</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> )</span><br/>

<a NAME="E2:80"/><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s2</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</font>  st <font color="Maroon" title="c7">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/>( <font color="Maroon" title="c7">s1</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> iff <font color="Olive" title="b1">s2</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies ( <font color="Maroon" title="c7">s1</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> iff <font color="Maroon" title="c8">s2</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> ) )</span><br/>



<b>assume </b><a NAME="E3:80"/><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">s1</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> iff <font color="Maroon" title="c8">s2</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> )</span><br/>

<a NAME="E4:80"/><i><font color="Green" title="E58">A4</font></i>: 
<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="#E1:80"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="#E3:80"><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/>
<b>thus </b><a NAME="E5:80"/>
( <font color="Maroon" title="c7">s1</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> implies <font color="Maroon" title="c8">s2</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">s2</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> implies <font color="Maroon" title="c7">s1</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_1"/>
<font color="Maroon" title="c7">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c7">s1</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="circuit2.html#D6" title="CIRCUIT2:def.6">CIRCUIT2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">s2</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a></span><br/>

<b>hence </b><a NAME="E2:80_1"/>
<font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c8">s2</font>
 <b>by </b><i><a class="txt" href="#E2:80"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="#E4:80"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="circtrm1.html#T55" target="_self" title="CIRCTRM1:th.55">Th55</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circuit2.html#D6" title="CIRCUIT2:def.6">CIRCUIT2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E6:80"/>
<font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c8">s2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="circuit2.html#D6" title="CIRCUIT2:def.6">CIRCUIT2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">s1</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a></span><br/>

<b>hence </b><a NAME="E7:80"/>
<font color="Maroon" title="c7">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c7">s1</font>
 <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="#E3:80"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="circtrm1.html#T55" target="_self" title="CIRCTRM1:th.55">Th55</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circuit2.html#D6" title="CIRCUIT2:def.6">CIRCUIT2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
