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

<b>set </b><font color="Maroon" title="c1">f1</font> = <font color="Maroon">F<sub>4</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c2">g1</font> = <font color="Maroon">F<sub>6</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c3">h1</font> = <font color="Maroon">F<sub>8</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c4">f2</font> = <font color="Maroon">F<sub>5</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c5">g2</font> = <font color="Maroon">F<sub>7</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c6">h2</font> = <font color="Maroon">F<sub>9</sub></font>();<br/>
<a NAME="E1:23_1"/><i><font color="Green" title="E12">A6</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">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">x</font>,0] )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E1:23"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/>
<a NAME="E2:23_1"/><i><font color="Green" title="E13">A7</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">F<sub>4</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="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</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="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</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">F<sub>4</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>1</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">F<sub>6</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>2</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>8</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>3</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#E3:23"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<a NAME="E3:23_1"/><i><font color="Green" title="E14">A8</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">F<sub>4</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="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</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="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</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>2</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>2</sub></font>[<font color="Maroon">F<sub>1</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">F<sub>2</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>3</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]
 
;<br/>
<a NAME="E4:23_1"/><i><font color="Green" title="E15">A9</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">F<sub>2</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">F<sub>1</sub></font>(<font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">n</font>)
 
<b>by </b><i><a class="txt" href="circcmb2.html#E5:23"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<a NAME="E5:23_1"/><i><font color="Green" title="E16">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">F<sub>4</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="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</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>2</sub></font>[<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">A</font>,<font color="Maroon">F<sub>8</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#E1:23_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="circcmb2.html#E2:23_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="circcmb2.html#E3:23_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="circcmb2.html#E4:23_1"><i><font color="Green" title="E15">A9</font></i></a>);<br/></i>
<a NAME="E6:23_1"/><i><font color="Green" title="E17">A11</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">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Olive" title="b2">A</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> 0 &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">x</font>,0] )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E1:23"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="circcmb2.html#E2:23"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<a NAME="E7:23_1"/><i><font color="Green" title="E18">A12</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">F<sub>5</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="b3">A</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="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</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>2</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>2</sub></font>[<font color="Maroon">F<sub>1</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">F<sub>2</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>3</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]
 
;<br/>
<a NAME="E8:23_1"/><i><font color="Green" title="E19">A13</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">F<sub>5</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="b3">A</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="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</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">F<sub>5</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>1</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">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>2</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>9</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>3</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#E4:23"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>
<a NAME="E9:23_1"/><i><font color="Green" title="E20">A14</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">F<sub>5</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="b3">A</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>2</sub></font>[<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">A</font>,<font color="Maroon">F<sub>9</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#E6:23_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="circcmb2.html#E8:23_1"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="circcmb2.html#E7:23_1"><i><font color="Green" title="E18">A12</font></i></a>, <a class="txt" href="circcmb2.html#E4:23_1"><i><font color="Green" title="E15">A9</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon">F<sub>8</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E10:23_1"/><i><font color="Green" title="E21">A15</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="txt" href="circcmb2.html#E2:23"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<div><i><font color="Green" title="E22">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green" title="E23">A17</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">n</font>]
 ;<br/><a NAME="E2:23_1_1"/>
 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">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="txt" href="circcmb2.html#E5:23_1"><i><font color="Green" title="E16">A10</font></i></a></i>;<br/><a NAME="E3:23_1_1"/><b>then </b><i><font color="Green" title="E24">A18</font></i>: 
<font color="Maroon">F<sub>8</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>8</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<font color="Maroon" title="c7">n</font>)
 <b>by </b><i><a class="txt" href="circcmb2.html#E3:23"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/><a NAME="E4:23_1_1"/>
 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">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Olive" title="b2">A</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="c7">n</font> )
 <b>by </b><i><a class="txt" href="circcmb2.html#E9:23_1"><i><font color="Green" title="E20">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E5:23_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">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#E4:23"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="circcmb2.html#E1:23_1_1"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="circcmb2.html#E3:23_1_1"><i><font color="Green" title="E24">A18</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:23_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="circcmb2.html#E10:23_1"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="circcmb2.html#E11:23_1"><i><font color="Green" title="E22">A16</font></i></a>);<br/></i>
<a NAME="E13:23_1"/><b>then </b><i><font color="Green" title="E23">A19</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">F<sub>8</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
;<br/>
<a NAME="E14:23_1"/><b>then </b><i><font color="Green" title="E24">A20</font></i>: 
<font color="Maroon">F<sub>8</sub></font>() <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">F<sub>9</sub></font>()
 
<b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <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> $1 );<br/>
<a NAME="E15:23_1"/><i><font color="Green" title="E25">A21</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<b>by </b><i><a class="txt" href="circcmb2.html#E2:23"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<div><i><font color="Green" title="E26">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:23_1_2"/><i><font color="Green" title="E27">A23</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c7">n</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c8">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> , <font color="Maroon" title="c9">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="Maroon" title="c8">S</font><b> such that </b><br/><a NAME="E3:23_1_2"/><i><font color="Green" title="E28">A24</font></i>: 
( <font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="txt" href="circcmb2.html#E5:23_1"><i><font color="Green" title="E16">A10</font></i></a></i>;<br/><i><font color="Green" title="E29">A25</font></i>: <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c8">S</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>8</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<font color="Maroon" title="c7">n</font>)
<b>by </b><i><a class="txt" href="circcmb2.html#E3:23"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="circcmb2.html#E3:23_1_2"><i><font color="Green" title="E28">A24</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="circcmb2.html#E4:23"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="circcmb2.html#E14:23_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="circcmb2.html#E1:23_1_2"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="circcmb2.html#E3:23_1_2"><i><font color="Green" title="E28">A24</font></i></a></i>
;<br/><font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c8">S</font>,<font color="Maroon" title="c9">A</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>8</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<font color="Maroon" title="c7">n</font>)
<b>by </b><i><a class="txt" href="circcmb2.html#E3:23"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="circcmb2.html#E3:23_1_2"><i><font color="Green" title="E28">A24</font></i></a></i>
<br/>.= 
<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="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="circcmb2.html#E4:23"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="circcmb2.html#E14:23_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="circcmb2.html#E1:23_1_2"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="circcmb2.html#E3:23_1_2"><i><font color="Green" title="E28">A24</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:23_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c7">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#E4:23_1_2"><i><font color="Green" title="E29">A25</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:23_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="circcmb2.html#E15:23_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="circcmb2.html#E16:23_1"><i><font color="Green" title="E26">A22</font></i></a>);<br/></i>
<a NAME="E18:23_1"/><b>then </b>
( ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</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">i</font> ) )
 
;<br/>
<b>hence </b><a NAME="E19:23_1"/>
( <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">F<sub>5</sub></font>() &amp; <font color="Maroon">F<sub>6</sub></font>() <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">F<sub>7</sub></font>() &amp; <font color="Maroon">F<sub>8</sub></font>() <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">F<sub>9</sub></font>() )
 <b>by </b><i><a class="txt" href="circcmb2.html#E13:23_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


</div>
