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<a NAME="E1:28_1"/><i><font color="Green" title="E11">A5</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>5</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>4</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#E3:28"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font>, <font color="Maroon" title="c3">h</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green" title="E12">A6</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() )
 <b>and </b><br/><a NAME="E4:28_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" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</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" title="c3">h</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) )
 <b>from </b><i><a class="ref" href="circcmb2.html#S12" target="_self" title="CIRCCMB2:sch.12">CIRCCMB2:sch 12</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">h1</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:28_1"/><i><font color="Green" title="E14">A8</font></i>: 
( <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c5">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() )
 <b>and </b><br/><a NAME="E7:28_1"/><i><font color="Green" title="E15">A9</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Maroon" title="c4">f1</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon" title="c5">h1</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>) )
 <b>by </b><i><a class="txt" href="circcmb2.html#E2:28"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<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" title="c5">h1</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" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E8:28_1"/><i><font color="Green" title="E16">A10</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="circcmb2.html#E3:28_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="circcmb2.html#E6:28_1"><i><font color="Green" title="E14">A8</font></i></a></i>;<br/>
<a NAME="E9:28_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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">x</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ] )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E6:28_1"><i><font color="Green" title="E14">A8</font></i></a></i>;<br/>
<a NAME="E10:28_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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h1</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>1</sub></font>[<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>),<font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>),<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
;<br/>
<a NAME="E11:28_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>   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>  st <br/>( <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</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>1</sub></font>[<font color="Olive" title="b2">S</font>,<font color="Maroon" title="c5">h1</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#S2" target="_self" title="CIRCCMB2:sch.2">CIRCCMB2:sch 2</a>(<a class="txt" href="circcmb2.html#E9:28_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="circcmb2.html#E7:28_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="circcmb2.html#E10:28_1"><i><font color="Green" title="E18">A12</font></i></a>);<br/></i>
<a NAME="E12:28_1"/><i><font color="Green" title="E20">A14</font></i>: 
 ex <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  ex <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b1">S</font> ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">x</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ] )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E3:28_1"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>
<a NAME="E13:28_1"/><i><font color="Green" title="E21">A15</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" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</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>4</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>5</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>7</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#K2" title="NAT_1:func.2">+</a> 1]
 
;<br/>
<a NAME="E14:28_1"/><i><font color="Green" title="E22">A16</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" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">A</font>,<font color="Maroon" title="c3">h</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#E12:28_1"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="circcmb2.html#E4:28_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="circcmb2.html#E13:28_1"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="circcmb2.html#E1:28_1"><i><font color="Green" title="E11">A5</font></i></a>);<br/></i>
<div><i><font color="Green" title="E23">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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="E1:28_1_1"/><i><font color="Green" title="E24">A18</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">n</font>]
 ;<br/><a NAME="E2:28_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>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="circcmb2.html#E11:28_1"><i><font color="Green" title="E19">A13</font></i></a></i>;<br/><a NAME="E3:28_1_1"/><b>then </b><i><font color="Green" title="E25">A19</font></i>: 
<font color="Maroon" title="c5">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c6">n</font>)
 <b>by </b><i><a class="txt" href="circcmb2.html#E7:28_1"><i><font color="Green" title="E15">A9</font></i></a></i>;<br/><a NAME="E4:28_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" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )
 <b>by </b><i><a class="txt" href="circcmb2.html#E14:28_1"><i><font color="Green" title="E22">A16</font></i></a></i>;<br/><b>hence </b><a NAME="E5:28_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="circcmb2.html#E4:28_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="circcmb2.html#E1:28_1_1"><i><font color="Green" title="E24">A18</font></i></a>, <a class="txt" href="circcmb2.html#E3:28_1_1"><i><font color="Green" title="E25">A19</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:28_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#E8:28_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="circcmb2.html#E15:28_1"><i><font color="Green" title="E23">A17</font></i></a>);<br/></i>
<a NAME="E17:28_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" title="c5">h1</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" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
;<br/>
<a NAME="E18:28_1"/><b>then </b><i><font color="Green" title="E24">A20</font></i>: 
<font color="Maroon" title="c5">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">h</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" title="c4">f1</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" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E19:28_1"/><i><font color="Green" title="E25">A21</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="circcmb2.html#E3:28_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="circcmb2.html#E6:28_1"><i><font color="Green" title="E14">A8</font></i></a></i>;<br/>
<div><i><font color="Green" title="E26">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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="E1:28_1_2"/><i><font color="Green" title="E27">A23</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c6">n</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c7">S</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> , <font color="Maroon" title="c8">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="c7">S</font><b> such that </b><br/><a NAME="E3:28_1_2"/><i><font color="Green" title="E28">A24</font></i>: 
( <font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )
 <b>by </b><i><a class="txt" href="circcmb2.html#E14:28_1"><i><font color="Green" title="E22">A16</font></i></a></i>;<br/><font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c6">n</font>)
<b>by </b><i><a class="txt" href="circcmb2.html#E7:28_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="circcmb2.html#E1:28_1_2"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="circcmb2.html#E3:28_1_2"><i><font color="Green" title="E28">A24</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="circcmb2.html#E4:28_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="circcmb2.html#E18:28_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="circcmb2.html#E3:28_1_2"><i><font color="Green" title="E28">A24</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:28_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E21:28_1"/><i><font color="Green" title="E27">A25</font></i>: 
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#E19:28_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="circcmb2.html#E20:28_1"><i><font color="Green" title="E26">A22</font></i></a>);<br/></i>
<a NAME="E22:28_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" title="c4">f1</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" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
;<br/>
<a NAME="E23:28_1"/><b>then </b><i><font color="Green" title="E28">A26</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c1">f</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>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="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> ex <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="circcomb.html#V1" title="CIRCCOMB:attr.1">unsplit</a> <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</a> <a href="circcomb.html#V3" title="CIRCCOMB:attr.3">gate`2isBoolean</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  ex <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Olive" title="b1">S</font> st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 );<br/>
<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="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">S<sub>5</sub></font>[$1,$2,$4];<br/>
<a NAME="E24:28_1"/><i><font color="Green" title="E29">A27</font></i>: 
 ex <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  ex <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b1">S</font> ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">x</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ] )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E3:28_1"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>
<a NAME="E25:28_1"/><i><font color="Green" title="E30">A28</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" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</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>6</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>6</sub></font>[<font color="Maroon">F<sub>4</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>5</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>7</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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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>let </b><font color="Maroon" title="c7">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c8">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c7">S</font>;<br/><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>assume </b><a NAME="E1:28_1_3"/><i><font color="Green" title="E31">A29</font></i>: 
( <font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c7">S</font>,<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c6">n</font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">S</font> = <font color="Maroon" title="c7">S</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="circcomb.html#V1" title="CIRCCOMB:attr.1">unsplit</a> <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</a> <a href="circcomb.html#V3" title="CIRCCOMB:attr.3">gate`2isBoolean</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">A</font> = <font color="Maroon" title="c8">A</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c10">S</font> <b>by </b><i><a class="txt" href="circcmb2.html#E1:28_1_3"><i><font color="Green" title="E31">A29</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">S1</font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c10">S</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c6">n</font>) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="circcomb.html#V1" title="CIRCCOMB:attr.1">unsplit</a> <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</a> <a href="circcomb.html#V3" title="CIRCCOMB:attr.3">gate`2isBoolean</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  <b>by </b><i><a class="txt" href="circcmb2.html#E1:28"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/>
<a NAME="E5:28_1_3"/>
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c10">S</font>,<font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c6">n</font>) is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c12">S1</font>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E4:28"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:28_1_3"/>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">S</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c6">n</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c7">S</font>,<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c6">n</font>),<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c6">n</font>),<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E26:28_1"/>
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" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">A</font>,<font color="Maroon" title="c3">h</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#E24:28_1"><i><font color="Green" title="E29">A27</font></i></a>, <a class="txt" href="circcmb2.html#E4:28_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="circcmb2.html#E25:28_1"><i><font color="Green" title="E30">A28</font></i></a>, <a class="txt" href="circcmb2.html#E1:28_1"><i><font color="Green" title="E11">A5</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c6">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="c7">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="c6">S</font><b> such that </b><br/><a NAME="E28:28_1"/><i><font color="Green" title="E31">A30</font></i>: 
( <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon">F<sub>8</sub></font>()] )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c7">A</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="circcmb2.html#E6:28_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="circcmb2.html#E23:28_1"><i><font color="Green" title="E28">A26</font></i></a>, <a class="txt" href="circcmb2.html#E28:28_1"><i><font color="Green" title="E31">A30</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">A</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c1">f</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c2">g</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c3">h</font>
;<br/>

<b>thus </b><a NAME="E30:28_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() &amp; ( 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" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">S</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</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" title="c3">h</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</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#E3:28_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="circcmb2.html#E4:28_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="circcmb2.html#E6:28_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="circcmb2.html#E21:28_1"><i><font color="Green" title="E27">A25</font></i></a>, <a class="txt" href="circcmb2.html#E28:28_1"><i><font color="Green" title="E31">A30</font></i></a></i>;<br/>


</div>
