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<span class="kw">let </span><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="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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c2">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="c1">G2</font>  st <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="circtrm1.html#R4" title="CIRCTRM1:pred.4">form_embedding_of</a> <font color="Olive" title="b3">C1</font>,<font color="Olive" title="b4">C2</font> holds <br/> for <font color="Olive" title="b5">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b4">C2</font><br/>  for <font color="Olive" title="b6">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b3">C1</font>  st <font color="Olive" title="b6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font> &amp; (  for <font color="Olive" title="b7">v</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">G1</font>  st <font color="Olive" title="b7">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="c2">G1</font> holds <br/><font color="Olive" title="b5">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">v</font> ) holds <br/> <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Olive" title="b6">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="Olive" title="b5">s2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c2">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="c1">G2</font>  st <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="Olive" title="b1">C1</font>,<font color="Olive" title="b2">C2</font> holds <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="b2">C2</font><br/>  for <font color="Olive" title="b4">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b1">C1</font>  st <font color="Olive" title="b4">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> &amp; (  for <font color="Olive" title="b5">v</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">G1</font>  st <font color="Olive" title="b5">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="c2">G1</font> holds <br/><font color="Olive" title="b3">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="Olive" title="b5">v</font> ) holds <br/> <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Olive" title="b4">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="Olive" title="b3">s2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span><br/><span class="kw">let </span><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="c2">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c1">G2</font>  st <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="Olive" title="b1">C2</font> holds <br/> for <font color="Olive" title="b2">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b1">C2</font><br/>  for <font color="Olive" title="b3">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font>  st <font color="Olive" title="b3">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> &amp; (  for <font color="Olive" title="b4">v</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">G1</font>  st <font color="Olive" title="b4">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="c2">G1</font> holds <br/><font color="Olive" title="b2">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="Olive" title="b4">v</font> ) holds <br/> <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Olive" title="b3">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="Olive" title="b2">s2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span><br/><span class="kw">let </span><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="c1">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies  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><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>  st <font color="Olive" title="b2">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> &amp; (  for <font color="Olive" title="b3">v</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">G1</font>  st <font color="Olive" title="b3">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="c2">G1</font> holds <br/><font color="Olive" title="b1">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="Olive" title="b3">v</font> ) holds <br/> <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Olive" title="b2">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="Olive" title="b1">s2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> )</span><br/>











<span class="kw">assume </span><a NAME="E1:69"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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><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>  st <font color="Olive" title="b2">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> &amp; (  for <font color="Olive" title="b3">v</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">G1</font>  st <font color="Olive" title="b3">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="c2">G1</font> holds <br/><font color="Olive" title="b1">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="Olive" title="b3">v</font> ) holds <br/> <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Olive" title="b2">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="Olive" title="b1">s2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span><br/>

<a NAME="E2:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#D12" target="_self" title="CIRCTRM1:def.12">Def12</a></span>;<br/>
<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>  st <font color="Olive" title="b1">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> &amp; (  for <font color="Olive" title="b2">v</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">G1</font>  st <font color="Olive" title="b2">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="c2">G1</font> holds <br/><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="Olive" title="b2">v</font> ) holds <br/> <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Olive" title="b1">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></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">G1</font>  st <font color="Olive" title="b1">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="c2">G1</font> holds <br/><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="Olive" title="b1">v</font> ) implies  <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> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:69"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:69"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
 for <font color="Olive" title="b1">v</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">G1</font>  st <font color="Olive" title="b1">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="c2">G1</font> holds <br/><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="Olive" title="b1">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">s29</font> = <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> as   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="circtrm1.html#T44" target="_self" title="CIRCTRM1:th.44">Th45</a></span>;<br/>
<a NAME="E6:69"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
 <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="c2">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="pua2mss1.html#D12" title="PUA2MSS1:def.12">PUA2MSS1:def 12</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">v</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c10">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="c2">G1</font> implies <font color="Maroon" title="c9">s29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font> ) &amp; ( <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">G1</font> implies <font color="Maroon" title="c9">s29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c10">v</font></span>)</span>,<font color="Maroon" title="c5">C1</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="circuit1.html#K3" title="CIRCUIT1:func.3">depends_on_in</a> <font color="Maroon" title="c8">s1</font></span>)</span> ) )</span><br/><a NAME="E1:69_1"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c9">s29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</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#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="c10">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">fv</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font> as   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="circtrm1.html#T30" target="_self" title="CIRCTRM1:th.30">Th31</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">G1</font> implies <font color="Maroon" title="c9">s29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c10">v</font></span>)</span>,<font color="Maroon" title="c5">C1</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="circuit1.html#K3" title="CIRCUIT1:func.3">depends_on_in</a> <font color="Maroon" title="c8">s1</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:69_1_1"/>
<font color="Maroon" title="c10">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="c2">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">s29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font></span><br/><a NAME="E2:69_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<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="c10">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E49">A4</font></span></a></span>;<br/><a NAME="E3:69_1_1"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s2</font>,1) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c7">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T14" title="FACIRC_1:th.14">FACIRC_1:14</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c9">s29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font> = 
<font color="Maroon" title="c7">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="c10">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E2:69_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="facirc_1.html#D9" title="FACIRC_1:def.9">FACIRC_1:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c8">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><a NAME="E4:69_1"/><span class="lab"><font color="Green" title="E52">A8</font></span>: 
<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#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">G1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="circtrm1.html#T32" target="_self" title="CIRCTRM1:th.32">Th33</a></span>;<br/><span class="kw">assume </span><a NAME="E5:69_1"/><span class="lab"><font color="Green" title="E53">A9</font></span>: 
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">s29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c10">v</font></span>)</span>,<font color="Maroon" title="c5">C1</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="circuit1.html#K3" title="CIRCUIT1:func.3">depends_on_in</a> <font color="Maroon" title="c8">s1</font></span>)</span></span><br/><a NAME="E6:69_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A10</font></span>: 
<font color="Maroon" title="c11">fv</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#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/><a NAME="E7:69_1"/><span class="lab"><font color="Green" title="E55">A11</font></span>: 
 <a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c11">fv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c10">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E5:69_1"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="ref" href="circtrm1.html#T33" target="_self" title="CIRCTRM1:th.33">Th34</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c9">s29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c11">fv</font></span>)</span>,<font color="Maroon" title="c6">C2</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c11">fv</font></span>)</span> <a href="circuit1.html#K3" title="CIRCUIT1:func.3">depends_on_in</a> <font color="Maroon" title="c7">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E4:69_1"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="txt" href="#E6:69_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="ref" href="circuit2.html#D5" title="CIRCUIT2:def.5">CIRCUIT2:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c10">v</font></span>)</span>,<font color="Maroon" title="c5">C1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c11">fv</font></span>)</span> <a href="circuit1.html#K3" title="CIRCUIT1:func.3">depends_on_in</a> <font color="Maroon" title="c7">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E7:69_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="circtrm1.html#T42" target="_self" title="CIRCTRM1:th.42">Th43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c10">v</font></span>)</span>,<font color="Maroon" title="c5">C1</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">action_at</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="circuit1.html#K3" title="CIRCUIT1:func.3">depends_on_in</a> <font color="Maroon" title="c8">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E7:69_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="circtrm1.html#T43" target="_self" title="CIRCTRM1:th.43">Th44</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:69"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circuit2.html#D5" title="CIRCUIT2:def.5">CIRCUIT2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
