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<div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,0</span>)</span> = 
<span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0</span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E2:110"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:110"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:110_2"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
 <a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="funct_7.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E4:110"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="funct_7.html#T24" target="_self">Th24</a>);<br/></i>
<b>hence </b><a NAME="E5:110"/>
<span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K9" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span>
 ;<br/>


</div>
