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<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="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/>( <font color="Olive" title="b1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Olive" title="b1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">G2</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="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/>( <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">G2</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="c1">G1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="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/>( <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">G2</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="c2">G2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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 ( <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">G2</font> ) )</span><br/>











<span class="kw">assume </span><a NAME="E1:73"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">G2</font> )</span><br/>

<a NAME="E2:73"/><span class="kw">then </span>
<font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#T36" target="_self" onmouseover="rs('circtrm1/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:73"/>
( <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#T28" target="_self" onmouseover="rs('circtrm1/T28')" onmouseout="rh()">Th28</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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