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<b>let </b><font color="Maroon" title="c1">IIG</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a> <a href="msafree2.html#V5" title="MSAFREE2:attr.5">monotonic</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">SCS</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">IIG</font><br/> 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="c1">IIG</font><br/> for <font color="Olive" title="b3">iv</font> being   <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Olive" title="b1">SCS</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="c1">IIG</font> holds <br/> <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font></span><br/><b>let </b><font color="Maroon" title="c2">SCS</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">IIG</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><br/> for <font color="Olive" title="b2">iv</font> being   <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</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="c1">IIG</font> holds <br/> <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font></span><br/><b>let </b><font color="Maroon" title="c3">v</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</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">iv</font> being   <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</font>  st <font color="Maroon" title="c3">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> holds <br/> <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c3">v</font>,<font color="Olive" title="b1">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span><br/><b>let </b><font color="Maroon" title="c4">iv</font> be    <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</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"> ( <font color="Maroon" title="c3">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  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> )</span><br/>







<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c3">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font><b> such that </b><br/><a NAME="E3:15"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">SCS</font>
 <b>and </b><br/><a NAME="E4:15"/><i><font color="Green" title="E16">A2</font></i>: 
 <a href="circuit2.html#K4" title="CIRCUIT2:func.4">IGTree</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" title="CIRCUIT2:func.3">Fix_inp_ext</a> <font color="Maroon" title="c4">iv</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="ref" href="circuit2.html#D3" target="_self" title="CIRCUIT2:def.3">Def3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">X</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font>;<br/>
<a NAME="E5:15"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
 
;<br/>
<a NAME="E6:15"/><i><font color="Green" title="E18">A4</font></i>: 
 <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">FreeOper</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16" title="MSAFREE:def.16">MSAFREE:def 16</a></i>;<br/>
<a NAME="E7:15"/><i><font color="Green" title="E19">A5</font></i>: 
<span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font>,<font color="Maroon" title="c3">v</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D13" title="MSAFREE:def.13">MSAFREE:def 13</a></i>;<br/>
<a NAME="E8:15"/>
 <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font>,<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font></span>)</span> : (  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b2">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">IIG</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="msafree.html#D12" title="MSAFREE:def.12">MSAFREE:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">a</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font></span>)</span><b> such that </b><br/><a NAME="E10:15"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <b>and </b><br/><a NAME="E11:15"/><i><font color="Green" title="E21">A7</font></i>: 
(  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">IIG</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> ) )
 <b>by </b><i><a class="txt" href="circuit2.html#E5:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="circuit2.html#E6:15"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="circuit2.html#E7:15"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:15"/><i><font color="Green" title="E22">A8</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="circuit2.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="circuit2.html#E11:15"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="msafree2.html#T2" title="MSAFREE2:th.2">MSAFREE2:2</a></i>;<br/>
<a NAME="E14:15"/><i><font color="Green" title="E23">A9</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c3">v</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font>
 
<b>by </b><i><a class="txt" href="circuit2.html#E10:15"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="circuit2.html#E13:15"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="msafree.html#D17" title="MSAFREE:def.17">MSAFREE:def 17</a></i>;<br/>
<a NAME="E15:15"/><i><font color="Green" title="E24">A10</font></i>: 
<span class="p1">(<span class="default"><a href="circuit2.html#K2" title="CIRCUIT2:func.2">Fix_inp</a> <font color="Maroon" title="c4">iv</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c3">v</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span>)</span>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit2.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="circuit2.html#D1" target="_self" title="CIRCUIT2:def.1">Def1</a></i>;<br/>
<a NAME="E16:15"/>
 <a href="circuit2.html#K2" title="CIRCUIT2:func.2">Fix_inp</a> <font color="Maroon" title="c4">iv</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="circuit2.html#K3" title="CIRCUIT2:func.3">Fix_inp_ext</a> <font color="Maroon" title="c4">iv</font>
 
<b>by </b><i><a class="ref" href="circuit2.html#D2" target="_self" title="CIRCUIT2:def.2">Def2</a></i>;<br/>
<a NAME="E17:15"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
<span class="p1">(<span class="default"><a href="circuit2.html#K2" title="CIRCUIT2:func.2">Fix_inp</a> <font color="Maroon" title="c4">iv</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K3" title="CIRCUIT2:func.3">Fix_inp_ext</a> <font color="Maroon" title="c4">iv</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">v</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E18:15"/>
<font color="Maroon" title="c5">e</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K2" title="CIRCUIT2:func.2">Fix_inp</a> <font color="Maroon" title="c4">iv</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">v</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit2.html#E14:15"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="circuit2.html#E15:15"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E19:15"/><b>then </b><i><font color="Green" title="E26">A12</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K2" title="CIRCUIT2:func.2">Fix_inp</a> <font color="Maroon" title="c4">iv</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" title="CIRCUIT2:func.3">Fix_inp_ext</a> <font color="Maroon" title="c4">iv</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="circuit2.html#E17:15"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>
<a NAME="E20:15"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c4">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
 
<b>by </b><i><a class="txt" href="circuit2.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="msafree2.html#D5" title="MSAFREE2:def.5">MSAFREE2:def 5</a></i>;<br/>
<a NAME="E21:15"/><b>then </b>
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span>)</span>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeGen</a> <font color="Maroon" title="c3">v</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D17" title="MSAFREE:def.17">MSAFREE:def 17</a></i>;<br/>
<a NAME="E22:15"/><b>then </b>
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span>)</span>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
 
;<br/>
<a NAME="E23:15"/><b>then </b><i><font color="Green" title="E28">A14</font></i>: 
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span>)</span>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">SCS</font>,<font color="Maroon" title="c3">v</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D13" title="MSAFREE:def.13">MSAFREE:def 13</a></i>;<br/>
<b>thus </b> <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">iv</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Eval</a> <font color="Maroon" title="c2">SCS</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span>)</span>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="circuit2.html#E4:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="circuit2.html#E14:15"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="circuit2.html#E15:15"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="circuit2.html#E19:15"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c4">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
<b>by </b><i><a class="txt" href="circuit2.html#E20:15"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="circuit2.html#E23:15"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="msafree2.html#D9" title="MSAFREE2:def.9">MSAFREE2:def 9</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
