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<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</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>  st <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> holds <br/>for <font color="Olive" title="b3">v1</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">S1</font>  st <font color="Olive" title="b3">v1</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">S1</font> holds <br/>for <font color="Olive" title="b4">v2</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">S2</font>  st <font color="Olive" title="b4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">v1</font> holds <br/> <a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b3">v1</font></span>)</span></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"> ( <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="c1">S1</font>,<font color="Maroon" title="c2">S2</font> implies for <font color="Olive" title="b1">v1</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">S1</font>  st <font color="Olive" title="b1">v1</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">S1</font> holds <br/>for <font color="Olive" title="b2">v2</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">S2</font>  st <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">v1</font> holds <br/> <a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b2">v2</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v1</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E35">A1</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="c1">S1</font>,<font color="Maroon" title="c2">S2</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">v1</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">S1</font>  st <font color="Olive" title="b1">v1</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">S1</font> holds <br/>for <font color="Olive" title="b2">v2</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">S2</font>  st <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">v1</font> holds <br/> <a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b2">v2</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v1</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c5">v1</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">S1</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">v1</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">S1</font> implies for <font color="Olive" title="b1">v2</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">S2</font>  st <font color="Olive" title="b1">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">v1</font> holds <br/> <a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v2</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c5">v1</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:54"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c5">v1</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">S1</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">v2</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">S2</font>  st <font color="Olive" title="b1">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">v1</font> holds <br/> <a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v2</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c5">v1</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c6">v2</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">S2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">v1</font> implies  <a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c6">v2</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c5">v1</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E3:54"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">v1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c6">v2</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c5">v1</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">g1</font> = <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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c5">v1</font></span>)</span> as   <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c2">S2</font> <b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="circtrm1.html#T32" target="_self" title="CIRCTRM1:th.32">Th32</a></i>;<br/>
<a NAME="E5:54"/><i><font color="Green" title="E38">A4</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S1</font> &amp;  <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="c1">S1</font> )
 
<b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/>
<a NAME="E6:54"/><i><font color="Green" title="E39">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:54"/><i><font color="Green" title="E40">A6</font></i>: 
<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#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c2">S2</font>
 
<b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="circtrm1.html#T33" target="_self" title="CIRCTRM1:th.33">Th33</a></i>;<br/>
<a NAME="E8:54"/><i><font color="Green" title="E41">A7</font></i>: 
<font color="Maroon" title="c6">v2</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#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">S1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:54"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E3:54"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E5:54"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c7">g1</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c5">v1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:54"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c5">v1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c5">v1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E6:54"><i><font color="Green" title="E39">A5</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="c6">v2</font>
<b>by </b><i><a class="txt" href="#E2:54"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E3:54"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="msafree2.html#D7" title="MSAFREE2:def.7">MSAFREE2:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E10:54"/>
 <a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c6">v2</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#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c5">v1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:54"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="#E8:54"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="msafree2.html#D7" title="MSAFREE2:def.7">MSAFREE2:def 7</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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