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<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">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="E2:9_1"/><i><font color="Green">E39</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">h</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>and </b><br/><a NAME="E3:9_1"/><i><font color="Green">E40</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">f</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">F<sub>3</sub></font>(<font color="Olive">S</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>4</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>) )
 <b>from </b><i><a class="ref" href="circcmb2.html#S1" target="_self">CIRCCMB2:sch 1</a>();<br/></i>
<a NAME="E4:9_1"/><i><font color="Green">E41</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">f</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>1</sub></font>[<font color="Olive">S</font>,<font color="Olive">x</font>,0] )
 
<b>by </b><i/>;<br/>
<a NAME="E5:9_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">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">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>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">F<sub>3</sub></font>(<font color="Olive">S</font>,<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Maroon">F<sub>4</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="E6:9_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>  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="Maroon">S<sub>1</sub></font>[<font color="Olive">S</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#S2" target="_self">CIRCCMB2:sch 2</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> <b> such that </b><br/><a NAME="E8:9_1"/><i><font color="Green">E45</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>5</sub></font>()
 ;<br/>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

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

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

<b>thus </b><a NAME="E9:9_1"/>
( <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>5</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">h</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( 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">f</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">F<sub>3</sub></font>(<font color="Olive">S</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>4</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>) ) ) )
 <b>by </b><i>, , </i>;<br/>


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