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

<b>consider </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:48_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>10</sub></font>(2) &amp; <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>10</sub></font>(2) )
 <b>and </b><br/><a NAME="E3:48_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>() )
 <b>and </b><br/><a NAME="E4:48_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/>for <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>5</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="circcomb.html#K4">+*</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>9</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) )
 <b>by </b><i><a class="txt" href="circcmb2.html#E4:48"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="msualg_1.html#V1">strict</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ex <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="circcomb.html#V4">gate`2=den</a> <a href="circcomb.html#V6">Boolean</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Olive">b<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="circuit1.html#NM3">State</a> of <font color="Olive">b<sub>2</sub></font> holds  <a href="facirc_1.html#K9">Following</a> <font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(0) <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span></span>)</span> is <a href="circuit2.html#V1">stable</a> ) );<br/>
<a NAME="E5:48_1"/><i><font color="Green">E31</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ex <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,0] )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E2:48"><i><font color="Green">E20</font></i></a>, <a class="txt" href="circcmb2.html#E3:48_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>( non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="msualg_1.html#L1">ManySortedSign</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>);<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>( non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">a<sub>1</sub></font>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">a<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>4</sub></font>) = <font color="Maroon">a<sub>2</sub></font> <a href="circcomb.html#K4">+*</a> <span class="p1">(<span class="default"><a href="circcmb2.html#K1" target="_self">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>4</sub></font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>4</sub></font>)</span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E6:48_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/>for <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>9</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) as  <a href="msualg_1.html#V4">strict</a> <a href="circcomb.html#V4">gate`2=den</a> <a href="circcomb.html#V6">Boolean</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) <b>by </b><i><a class="txt" href="circcmb2.html#E1:48"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E2:48_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="circcmb2.html#K1" target="_self">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>)
 
<b>by </b><i><a class="ref" href="circcmb2.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:48_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>9</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) ) )
 <b>by </b><i><a class="txt" href="circcmb2.html#E4:48_1"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:48_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/>for <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">F<sub>9</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>),<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:48_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>given </b><font color="Maroon">c<sub>7</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="msualg_1.html#V1">strict</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">c<sub>8</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="circcomb.html#V4">gate`2=den</a> <a href="circcomb.html#V6">Boolean</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><a NAME="E2:48_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E3:48_1_2"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>8</sub></font> holds  <a href="facirc_1.html#K9">Following</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(0) <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span></span>)</span> is <a href="circuit2.html#V1">stable</a>
 ;<br/>

<b>thus </b><a NAME="E4:48_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">c<sub>5</sub></font> <a href="circcomb.html#K4">+*</a> <span class="p1">(<span class="default"><a href="circcmb2.html#K1" target="_self">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>)</span>)</span>,<font color="Maroon">F<sub>9</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</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">c<sub>7</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>)
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) as  <a href="msualg_1.html#V4">strict</a> <a href="circcomb.html#V4">gate`2=den</a> <a href="circcomb.html#V6">Boolean</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) <b>by </b><i><a class="txt" href="circcmb2.html#E1:48"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E2:48_1_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="circcmb2.html#K1" target="_self">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>)
 
<b>by </b><i><a class="ref" href="circcmb2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:48_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="circcomb.html#K4">+*</a> <span class="p1">(<span class="default"><a href="circcmb2.html#K1" target="_self">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>)</span>)</span>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E2:48_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="circcmb2.html#D1" target="_self">Def1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="circcomb.html#K4">+*</a> <span class="p1">(<span class="default"><a href="circcmb2.html#K1" target="_self">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>)</span>)</span> as  <a href="msualg_1.html#V4">strict</a> <a href="circcomb.html#V4">gate`2=den</a> <a href="circcomb.html#V6">Boolean</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">c<sub>7</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) ;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E5:48_1_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="circcomb.html#K4">+*</a> <span class="p1">(<span class="default"><a href="circcmb2.html#K1" target="_self">MSAlg</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="circcmb2.html#E2:48_1_2"><i><font color="Green">E35</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1)
 <b>by </b><i><a class="txt" href="circcmb2.html#E9:48"><i><font color="Green">E27</font></i></a>, <a class="txt" href="circcmb2.html#E2:48_1_2"><i><font color="Green">E35</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>10</sub></font>;<br/>

<a NAME="E7:48_1_2_1"/><i><font color="Green">E38</font></i>: 
 <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>1</sub></font>() is   <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E5:48"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E8:48_1_2_1"/><i><font color="Green">E39</font></i>: 
not  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>1</sub></font>() is <a href="facirc_1.html#V2">with_pair</a>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E5:48"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E9:48_1_2_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">H<sub>4</sub></font>(0) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">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">E24</font></i></a>, <a class="txt" href="circcmb2.html#E3:48_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E10:48_1_2_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</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">E29</font></i></a></i>;<br/>
<a NAME="E11:48_1_2_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/>for <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>) is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>)
 
;<br/>
<a NAME="E12:48_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>9</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>) )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S15" target="_self">CIRCCMB2:sch 15</a>(<a class="txt" href="circcmb2.html#E10:48_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="circcmb2.html#E6:48_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="circcmb2.html#E11:48_1_2_1"><i><font color="Green">E42</font></i></a>);<br/></i>
<a NAME="E13:48_1_2_1"/><b>then </b><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>9</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Maroon">F<sub>9</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>) )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E9:48"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E14:48_1_2_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>) is   <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E7:48"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E15:48_1_2_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> not <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>)</span>)</span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> is <a href="facirc_1.html#V2">with_pair</a>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E8:48"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E16:48_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1) &amp;  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">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">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font>)</span>)</span> <a href="xboole_0.html#K4">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <font color="Olive">b<sub>2</sub></font> is   <a href="relat_1.html#NM1">Relation</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">b<sub>2</sub></font> is <a href="facirc_1.html#V2">with_pair</a> )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S10" target="_self">CIRCCMB2:sch 10</a>(<a class="txt" href="circcmb2.html#E7:48_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="circcmb2.html#E8:48_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="circcmb2.html#E9:48_1_2_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="circcmb2.html#E14:48_1_2_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="circcmb2.html#E15:48_1_2_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="circcmb2.html#E13:48_1_2_1"><i><font color="Green">E43</font></i></a>);<br/></i>
<a NAME="E17:48_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xboole_0.html#K4">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span><a href="tarski.html#K1">}</a></span></span>)</span> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <font color="Olive">b<sub>1</sub></font> is   <a href="relat_1.html#NM1">Relation</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">b<sub>1</sub></font> is <a href="facirc_1.html#V2">with_pair</a> ) )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E6:48_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="circcmb2.html#E1:48_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="circcmb2.html#E2:48_1_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E18:48_1_2_1"/><b>then </b>
( not  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>7</sub></font> is <a href="facirc_1.html#V2">with_pair</a> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>) is   <a href="relat_1.html#NM1">Relation</a> )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E7:48"><i><font color="Green">E25</font></i></a>, <a class="txt" href="circcmb2.html#E1:48_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="circcmb2.html#E2:48_1_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E19:48_1_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>)
 
<b>by </b><i><a class="ref" href="facirc_1.html#T5">FACIRC_1:5</a></i>;<br/>
<a NAME="E20:48_1_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="circcomb.html#R2">tolerates</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="circcomb.html#T68">CIRCCOMB:68</a></i>;<br/>
<a NAME="E21:48_1_2_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>9</sub></font> holds  <a href="facirc_1.html#K9">Following</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>10</sub></font>(1) is <a href="circuit2.html#V1">stable</a>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E3:48"><i><font color="Green">E21</font></i></a>, <a class="txt" href="circcmb2.html#E2:48_1_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E22:48_1_2_1"/>
<font color="Maroon">F<sub>10</sub></font>(0) <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(0) <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">F<sub>10</sub></font>(1)
 
;<br/>
<b>hence </b><a NAME="E23:48_1_2_1"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(0) <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span></span>)</span> is <a href="circuit2.html#V1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb2.html#E2:48_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="circcmb2.html#E3:48_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="circcmb2.html#E2:48_1_2_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="circcmb2.html#E19:48_1_2_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="circcmb2.html#E20:48_1_2_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="circcmb2.html#E21:48_1_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="circcmb2.html#T21" target="_self">Th21</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E8:48_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/>for <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>) is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>)
 
;<br/>
<a NAME="E9:48_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ex <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">b<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S13" target="_self">CIRCCMB2:sch 13</a>(<a class="txt" href="circcmb2.html#E5:48_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="circcmb2.html#E6:48_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="circcmb2.html#E7:48_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circcmb2.html#E8:48_1"><i><font color="Green">E34</font></i></a>);<br/></i>
<a NAME="E10:48_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ex <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">b<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>10</sub></font>(2) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>10</sub></font>(2) &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">F<sub>7</sub></font>() <a href="funct_1.html#K1">.</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">b<sub>1</sub></font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">F<sub>4</sub></font>() holds  <a href="facirc_1.html#K9">Following</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(0) <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>10</sub></font>(2) <a href="nat_1.html#K2">*</a> <font color="Maroon">F<sub>10</sub></font>(1)</span>)</span></span>)</span> is <a href="circuit2.html#V1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb2.html#E2:48_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div>
