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<div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</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>4</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>2</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>4</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>5</sub></font>(<font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>4</sub></font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>4</sub></font>)</span>)</span>;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">g</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">h</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:30_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">f</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">g</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">h</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() )
 <b>and </b><br/><a NAME="E3:30_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/> for <font color="Olive">A</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">S</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">S</font>,<font color="Olive">x</font>,<font color="Olive">n</font>) &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</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">S</font>,<font color="Olive">A</font>,<font color="Olive">x</font>,<font color="Olive">n</font>) &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>) )
 <b>from </b><i><a class="ref" href="circcmb2.html#S12" target="_self">CIRCCMB2:sch 12</a>();<br/></i>
<b>consider </b><font color="Maroon">f1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">h1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:30_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon">f1</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">h1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() )
 <b>and </b><br/><a NAME="E6:30_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/>( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">S</font>,<font color="Olive">x</font>,<font color="Olive">n</font>) &amp; <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>) )
 <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E7:30_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:30_1"/><i><font color="Green">E49</font></i>: 
 ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>,<font color="Olive">x</font>,0] )
 
<b>by </b><i/>;<br/>
<a NAME="E9:30_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>,<font color="Olive">x</font>,<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">S</font>,<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Maroon">F<sub>7</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E10:30_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>,<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Olive">n</font>] )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S2" target="_self">CIRCCMB2:sch 2</a>(, , );<br/></i>
<a NAME="E11:30_1"/><i><font color="Green">E78</font></i>: 
 ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  ex <font color="Olive">A</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">S</font> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">x</font>,0] )
 
<b>by </b><i/>;<br/>
<a NAME="E12:30_1"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/> for <font color="Olive">A</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">S</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">x</font>,<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">S</font>,<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Maroon">F<sub>7</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E13:30_1"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/> for <font color="Olive">A</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">S</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">x</font>,<font color="Olive">n</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>1</sub></font>(<font color="Olive">S</font>,<font color="Olive">x</font>,<font color="Olive">n</font>)
 
;<br/>
<a NAME="E14:30_1"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  ex <font color="Olive">A</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">S</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Olive">n</font>] )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S13" target="_self">CIRCCMB2:sch 13</a>(, , , );<br/></i>
<div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:30_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:30_1_1"/>
 ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E3:30_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font>)
 <b>by </b><i/>;<br/><a NAME="E4:30_1_1"/>
 ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  ex <font color="Olive">A</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">S</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:30_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E16:30_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E17:30_1"/><b>then </b>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
;<br/>
<a NAME="E18:30_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">h1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E19:30_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:30_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>consider </b><font color="Maroon">S</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">A</font> being  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:30_1_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">S</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font>)
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E5:30_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E21:30_1"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E22:30_1"/><b>then </b>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
;<br/>
<a NAME="E23:30_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">f1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies  ex <font color="Olive">S</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">A</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">S</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>6</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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>4</sub></font>];<br/>
<a NAME="E24:30_1"/><i><font color="Green">E112</font></i>: 
 ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  ex <font color="Olive">A</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">S</font> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">x</font>,0] )
 
<b>by </b><i/>;<br/>
<a NAME="E25:30_1"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/> for <font color="Olive">A</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">S</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">x</font>,<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">S</font>,<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Maroon">F<sub>7</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">S</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">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:30_1_3"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">S</font>,<font color="Maroon">A</font>,<font color="Maroon">n</font>] &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_3_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_3_1_1"><b>suppose </b></a><a NAME="E1:30_1_3_1_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">x</font>,0) 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>4</sub></font>(<font color="Maroon">x</font>,0) <b>by </b><i/>;<br/><a NAME="E3:30_1_3_1_1"/>
<font color="Maroon">F<sub>3</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>5</sub></font>(<font color="Maroon">x</font>,0),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">x</font>,0)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="circcomb.html#K4">+*</a> <font color="Maroon">A2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:30_1_3_1_1"/>
 ex <font color="Olive">S</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">A</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">S</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">n</font>) &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">S</font>,<font color="Maroon">A</font>,<font color="Maroon">x</font>,<font color="Maroon">n</font>) )
 <b>by </b><i>, , <a class="ref" href="circcmb2.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_3_1_2"><b>suppose </b></a><a NAME="E1:30_1_3_1_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as  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>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</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">S</font> <b>by </b><i>, <a class="ref" href="circcmb2.html#T1" target="_self">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">n</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>4</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">n</font>) <b>by </b><i/>;<br/><a NAME="E5:30_1_3_1_2"/>
<font color="Maroon">A</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>5</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">n</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">n</font>)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">A'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:30_1_3_1_2"/>
 ex <font color="Olive">S</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">A</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">S</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">n</font>) &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">S</font>,<font color="Maroon">A</font>,<font color="Maroon">x</font>,<font color="Maroon">n</font>) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E26:30_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  ex <font color="Olive">A</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">S</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Olive">n</font>] )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S13" target="_self">CIRCCMB2:sch 13</a>(, , , );<br/></i>
<b>then consider </b><font color="Maroon">S</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">A</font> being  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E28:30_1"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">S</font>,<font color="Maroon">A</font>,<font color="Maroon">F<sub>8</sub></font>()] )
 ;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</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>2</sub></font>() <b>by </b><i>, , , </i>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E30:30_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon">f</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">g</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">h</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() )
 <b>by </b><i>, , , </i>;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">S</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">A1</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A2</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">n</font>);<br/>









<b>assume </b><a NAME="E31:30_1"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">n</font>) )
 ;<br/>

<a NAME="E32:30_1"/>
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="circcmb2.html#K1" target="_self">MSAlg</a> <font color="Maroon">A2</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">n</font>)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E33:30_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">n</font>) &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">n</font>) )
 <b>by </b><i>, <a class="ref" href="circcmb2.html#T1" target="_self">Th1</a></i>;<br/>


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