<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:48_1"/><i><font color="Green" title="E27">A10</font></i>: 
( <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>10</sub></font>(2) &amp; <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>10</sub></font>(2) )
 <b>and </b><br/><a NAME="E3:48_1"/><i><font color="Green" title="E28">A11</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>() )
 <b>and </b><br/><a NAME="E4:48_1"/><i><font color="Green" title="E29">A12</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/> for <font color="Olive" title="b3">A1</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b2">S</font><br/> for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b5">A2</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>)  st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b5">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) holds <br/>( <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A1</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Olive" title="b5">A2</font> &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) )
 <b>by </b><i><a class="txt" href="circcmb2.html#E4:48"><i><font color="Green" title="E21">A4</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b> ex <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="circcomb.html#V1" title="CIRCCOMB:attr.1">unsplit</a> <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</a> <a href="circcomb.html#V3" title="CIRCCOMB:attr.3">gate`2isBoolean</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  ex <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Olive" title="b1">S</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $4 &amp; ( for <font color="Olive" title="b3">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b2">A</font> holds   <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b3">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">$4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span></span>)</span> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> ) );<br/>
<a NAME="E5:48_1"/><i><font color="Green" title="E30">A13</font></i>: 
 ex <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  ex <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b1">S</font> ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">x</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ] )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E2:48"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="circcmb2.html#E3:48_1"><i><font color="Green" title="E28">A11</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>( non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  = $1 <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">F<sub>5</sub></font>($2,$3);<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>( non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> , <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of $1,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of $1 <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">F<sub>5</sub></font>($3,$4) = $2 <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <span class="p1">(<span class="default"><a href="circcmb2.html#K1" title="CIRCCMB2:func.1">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>($3,$4),<font color="Maroon">F<sub>5</sub></font>($3,$4)</span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E6:48_1"/><i><font color="Green" title="E31">A14</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/> for <font color="Olive" title="b3">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b2">S</font><br/> for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c4">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c5">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c4">S</font>;<br/><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>reconsider </b><font color="Maroon" title="c7">A2</font> = <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) <b>by </b><i><a class="txt" href="circcmb2.html#E1:48"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<a NAME="E2:48_1_1"/>
<font color="Maroon" title="c7">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcmb2.html#K1" title="CIRCCMB2:func.1">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>)
 
<b>by </b><i><a class="ref" href="circcmb2.html#D1" target="_self" title="CIRCCMB2:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:48_1_1"/>
( <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> implies ( <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) ) )
 <b>by </b><i><a class="txt" href="circcmb2.html#E4:48_1"><i><font color="Green" title="E29">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:48_1"/><i><font color="Green" title="E32">A15</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/> for <font color="Olive" title="b3">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b2">S</font><br/> for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>),<font color="Maroon">F<sub>9</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>),<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c4">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c5">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c4">S</font>;<br/><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>assume </b><a NAME="E1:48_1_2"/><i><font color="Green" title="E33">A16</font></i>: 
( <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> )
 ;<br/>

<b>given </b><font color="Maroon" title="c7">S'</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="circcomb.html#V1" title="CIRCCOMB:attr.1">unsplit</a> <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</a> <a href="circcomb.html#V3" title="CIRCCOMB:attr.3">gate`2isBoolean</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> , <font color="Maroon" title="c8">A'</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c7">S'</font><b> such that </b><a NAME="E3:48_1_2"/><i><font color="Green" title="E34">A17</font></i>: 
( <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S'</font> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A'</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> )
 <b>and </b><br/><a NAME="E4:48_1_2"/><i><font color="Green" title="E35">A18</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c8">A'</font> holds   <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span></span>)</span> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>
 ;<br/>

<b>thus </b><a NAME="E5:48_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">S</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>),<font color="Maroon" title="c5">A</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <span class="p1">(<span class="default"><a href="circcmb2.html#K1" title="CIRCCMB2:func.1">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon">F<sub>9</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>),<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c7">S'</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>)
;<br/>

<b>reconsider </b><font color="Maroon" title="c9">A2</font> = <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) <b>by </b><i><a class="txt" href="circcmb2.html#E1:48"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<a NAME="E2:48_1_2_1"/><i><font color="Green" title="E36">A19</font></i>: 
<font color="Maroon" title="c9">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcmb2.html#K1" title="CIRCCMB2:func.1">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>)
 
<b>by </b><i><a class="ref" href="circcmb2.html#D1" target="_self" title="CIRCCMB2:def.1">Def1</a></i>;<br/>
<a NAME="E3:48_1_2_1"/>
<font color="Maroon" title="c8">A'</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c9">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <span class="p1">(<span class="default"><a href="circcmb2.html#K1" title="CIRCCMB2:func.1">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E3:48_1_2"><i><font color="Green" title="E34">A17</font></i></a>, <a class="ref" href="circcmb2.html#D1" target="_self" title="CIRCCMB2:def.1">Def1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">AA</font> = <font color="Maroon" title="c5">A</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <span class="p1">(<span class="default"><a href="circcmb2.html#K1" title="CIRCCMB2:func.1">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>)</span>)</span> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c7">S'</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) ;<br/>
<b>take </b>
<font color="Maroon" title="c10">AA</font>
;<br/>

<b>thus </b><a NAME="E5:48_1_2_1"/>
( <font color="Maroon" title="c7">S'</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) &amp; <font color="Maroon" title="c5">A</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <span class="p1">(<span class="default"><a href="circcmb2.html#K1" title="CIRCCMB2:func.1">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">AA</font> )
 <b>by </b><i><a class="txt" href="circcmb2.html#E3:48_1_2"><i><font color="Green" title="E34">A17</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:48_1_2_1"/>
<font color="Maroon">F<sub>9</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <b>by </b><i><a class="txt" href="circcmb2.html#E9:48"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="circcmb2.html#E3:48_1_2"><i><font color="Green" title="E34">A17</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c11">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c10">AA</font>;<br/>

<a NAME="E7:48_1_2_1"/><i><font color="Green" title="E37">A20</font></i>: 
 <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">F<sub>1</sub></font>() is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E5:48"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E8:48_1_2_1"/><i><font color="Green" title="E38">A21</font></i>: 
not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">F<sub>1</sub></font>() is <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E5:48"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E9:48_1_2_1"/><i><font color="Green" title="E39">A22</font></i>: 
( <font color="Maroon">H<sub>4</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E6:48"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="circcmb2.html#E3:48_1"><i><font color="Green" title="E28">A11</font></i></a></i>;<br/>
<a NAME="E10:48_1_2_1"/><i><font color="Green" title="E40">A23</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E3:48_1"><i><font color="Green" title="E28">A11</font></i></a></i>;<br/>
<a NAME="E11:48_1_2_1"/><i><font color="Green" title="E41">A24</font></i>: 
for <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/> for <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">n</font>) is  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">n</font>)
 
;<br/>
<a NAME="E12:48_1_2_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>(<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>) )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S15" target="_self" title="CIRCCMB2:sch.15">CIRCCMB2:sch 15</a>(<a class="txt" href="circcmb2.html#E10:48_1_2_1"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="circcmb2.html#E6:48_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="txt" href="circcmb2.html#E11:48_1_2_1"><i><font color="Green" title="E41">A24</font></i></a>);<br/></i>
<a NAME="E13:48_1_2_1"/><b>then </b><i><font color="Green" title="E42">A25</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>(<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b3">x</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">F<sub>5</sub></font>(<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon">F<sub>9</sub></font>(<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</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">F<sub>5</sub></font>(<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>) )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E9:48"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/>
<a NAME="E14:48_1_2_1"/><i><font color="Green" title="E43">A26</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>) is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E7:48"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/>
<a NAME="E15:48_1_2_1"/><i><font color="Green" title="E44">A27</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/> not <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E8:48"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/>
<a NAME="E16:48_1_2_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b2">S1</font>, <font color="Olive" title="b3">S2</font> being 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="circcomb.html#V1" title="CIRCCOMB:attr.1">unsplit</a> <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</a> <a href="circcomb.html#V3" title="CIRCCOMB:attr.3">gate`2isBoolean</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  st <br/>( <font color="Olive" title="b2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b3">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) &amp;  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Olive" title="b3">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Olive" title="b2">S1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp;  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Olive" title="b2">S1</font> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> &amp; not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Olive" title="b2">S1</font> is <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a> )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S10" target="_self" title="CIRCCMB2:sch.10">CIRCCMB2:sch 10</a>(<a class="txt" href="circcmb2.html#E7:48_1_2_1"><i><font color="Green" title="E37">A20</font></i></a>, <a class="txt" href="circcmb2.html#E8:48_1_2_1"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="circcmb2.html#E9:48_1_2_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="txt" href="circcmb2.html#E14:48_1_2_1"><i><font color="Green" title="E43">A26</font></i></a>, <a class="txt" href="circcmb2.html#E15:48_1_2_1"><i><font color="Green" title="E44">A27</font></i></a>, <a class="txt" href="circcmb2.html#E13:48_1_2_1"><i><font color="Green" title="E42">A25</font></i></a>);<br/></i>
<a NAME="E17:48_1_2_1"/><b>then </b>
( <font color="Maroon" title="c7">S'</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  ex <font color="Olive" title="b1">S1</font>, <font color="Olive" title="b2">S2</font> being 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="circcomb.html#V1" title="CIRCCOMB:attr.1">unsplit</a> <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</a> <a href="circcomb.html#V3" title="CIRCCOMB:attr.3">gate`2isBoolean</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  st <br/>( <font color="Olive" title="b1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Olive" title="b2">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Olive" title="b2">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Olive" title="b1">S1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>),<font color="Maroon" title="c3">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp;  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Olive" title="b1">S1</font> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> &amp; not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Olive" title="b1">S1</font> is <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a> ) )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E6:48_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="txt" href="circcmb2.html#E1:48_1_2"><i><font color="Green" title="E33">A16</font></i></a>, <a class="txt" href="circcmb2.html#E3:48_1_2"><i><font color="Green" title="E34">A17</font></i></a></i>;<br/>
<a NAME="E18:48_1_2_1"/><b>then </b><i><font color="Green" title="E45">A28</font></i>: 
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c7">S'</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">n</font>)
 
<b>by </b><i><a class="txt" href="circcmb2.html#E7:48"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="circcmb2.html#E1:48_1_2"><i><font color="Green" title="E33">A16</font></i></a>, <a class="txt" href="circcmb2.html#E3:48_1_2"><i><font color="Green" title="E34">A17</font></i></a>, <a class="ref" href="facirc_1.html#T5" title="FACIRC_1:th.5">FACIRC_1:5</a></i>;<br/>
<a NAME="E19:48_1_2_1"/><i><font color="Green" title="E46">A29</font></i>: 
<font color="Maroon" title="c8">A'</font> <a href="circcomb.html#R2" title="CIRCCOMB:pred.2">tolerates</a> <font color="Maroon" title="c9">A2</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#T68" title="CIRCCOMB:th.68">CIRCCOMB:68</a></i>;<br/>
<a NAME="E20:48_1_2_1"/><i><font color="Green" title="E47">A30</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c9">A2</font> holds   <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b1">s</font>,<font color="Maroon">F<sub>10</sub></font>(1) is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E3:48"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="circcmb2.html#E3:48_1_2"><i><font color="Green" title="E34">A17</font></i></a></i>;<br/>
<a NAME="E21:48_1_2_1"/>
<font color="Maroon">F<sub>10</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon">F<sub>10</sub></font>(1)
 
;<br/>
<b>hence </b><a NAME="E22:48_1_2_1"/>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span></span>)</span> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb2.html#E3:48_1_2"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="circcmb2.html#E4:48_1_2"><i><font color="Green" title="E35">A18</font></i></a>, <a class="txt" href="circcmb2.html#E2:48_1_2_1"><i><font color="Green" title="E36">A19</font></i></a>, <a class="txt" href="circcmb2.html#E18:48_1_2_1"><i><font color="Green" title="E45">A28</font></i></a>, <a class="txt" href="circcmb2.html#E19:48_1_2_1"><i><font color="Green" title="E46">A29</font></i></a>, <a class="txt" href="circcmb2.html#E20:48_1_2_1"><i><font color="Green" title="E47">A30</font></i></a>, <a class="ref" href="circcmb2.html#T21" target="_self" title="CIRCCMB2:th.21">Th21</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E8:48_1"/><i><font color="Green" title="E33">A31</font></i>: 
for <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/> for <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">n</font>) is  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">n</font>)
 
;<br/>
<a NAME="E9:48_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b2">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  ex <font color="Olive" title="b3">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b2">S</font> st <br/>( <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">A</font>,<font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b1">n</font>] )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S13" target="_self" title="CIRCCMB2:sch.13">CIRCCMB2:sch 13</a>(<a class="txt" href="circcmb2.html#E5:48_1"><i><font color="Green" title="E30">A13</font></i></a>, <a class="txt" href="circcmb2.html#E6:48_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="txt" href="circcmb2.html#E7:48_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="circcmb2.html#E8:48_1"><i><font color="Green" title="E33">A31</font></i></a>);<br/></i>
<a NAME="E10:48_1"/><b>then </b>
 ex <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  ex <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b1">S</font> st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>10</sub></font>(2) &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>10</sub></font>(2) &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">A</font>,<font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>10</sub></font>(2),<font color="Maroon">F<sub>10</sub></font>(2)] )
 
;<br/>
<b>hence </b><a NAME="E11:48_1"/>
for <font color="Olive" title="b1">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon">F<sub>4</sub></font>() holds   <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(2) <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span></span>)</span> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb2.html#E2:48_1"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>


</div>
