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<b>let </b><font color="Maroon" title="c4">it1</font>, <font color="Maroon" title="c5">it2</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</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">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font> holds <br/> ( ( <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="c1">IIG</font> implies <font color="Maroon" title="c4">it1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> ) &amp; ( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font> implies <font color="Maroon" title="c4">it1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v</font></span>)</span>,<font color="Maroon" title="c2">SCS</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="circuit1.html#K3" title="CIRCUIT1:func.3">depends_on_in</a> <font color="Maroon" title="c3">s</font></span>)</span> ) ) ) &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="c1">IIG</font> holds <br/> ( ( <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="c1">IIG</font> implies <font color="Maroon" title="c5">it2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> ) &amp; ( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font> implies <font color="Maroon" title="c5">it2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v</font></span>)</span>,<font color="Maroon" title="c2">SCS</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="circuit1.html#K3" title="CIRCUIT1:func.3">depends_on_in</a> <font color="Maroon" title="c3">s</font></span>)</span> ) ) ) implies <font color="Maroon" title="c4">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">it2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_1_2"/><i><font color="Green" title="E16">A7</font></i>: 
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="c1">IIG</font> holds <br/> ( ( <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="c1">IIG</font> implies <font color="Maroon" title="c4">it1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> ) &amp; ( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font> implies <font color="Maroon" title="c4">it1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v</font></span>)</span>,<font color="Maroon" title="c2">SCS</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="circuit1.html#K3" title="CIRCUIT1:func.3">depends_on_in</a> <font color="Maroon" title="c3">s</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:17_1_2"/><i><font color="Green" title="E17">A8</font></i>: 
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="c1">IIG</font> holds <br/> ( ( <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="c1">IIG</font> implies <font color="Maroon" title="c5">it2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> ) &amp; ( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font> implies <font color="Maroon" title="c5">it2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v</font></span>)</span>,<font color="Maroon" title="c2">SCS</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="circuit1.html#K3" title="CIRCUIT1:func.3">depends_on_in</a> <font color="Maroon" title="c3">s</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">it2</font></span><br/>

<a NAME="E3:17_1_2"/><i><font color="Green" title="E18">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">it1</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="c1">IIG</font>
 
<b>by </b><i><a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></i>;<br/>
<a NAME="E4:17_1_2"/><i><font color="Green" title="E19">A10</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">it2</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="c1">IIG</font>
 
<b>by </b><i><a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></i>;<br/>
<b>assume </b><a NAME="E5:17_1_2"/>
<font color="Maroon" title="c4">it1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">it2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:17_1_2"/><i><font color="Green" title="E20">A11</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">it1</font> &amp; <font color="Maroon" title="c4">it1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">it2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="circuit2.html#E3:17_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="circuit2.html#E4:17_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">v</font> = <font color="Maroon" title="c6">x</font> as   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font> <b>by </b><i><a class="txt" href="circuit2.html#E7:17_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></i>;<br/>
<a NAME="E9:17_1_2"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit2.html#E3:17_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="circuit2.html#E7:17_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E10:17_1_2"/><b>then </b><i><font color="Green" title="E21">A12</font></i>: 
( <font color="Maroon" title="c7">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="c1">IIG</font> or <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:17_1_2"/>
( ( <font color="Maroon" title="c7">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="c1">IIG</font> implies <font color="Maroon" title="c4">it1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">v</font> ) &amp; ( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font> implies <font color="Maroon" title="c4">it1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c7">v</font></span>)</span>,<font color="Maroon" title="c2">SCS</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="circuit1.html#K3" title="CIRCUIT1:func.3">depends_on_in</a> <font color="Maroon" title="c3">s</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="circuit2.html#E1:17_1_2"><i><font color="Green" title="E16">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:17_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="circuit2.html#E2:17_1_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="circuit2.html#E7:17_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="circuit2.html#E10:17_1_2"><i><font color="Green" title="E21">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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