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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default">0 <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">m</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font>
<b>by </b><i/>
;<br/>
<a NAME="E2:44"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:44"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:44_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">R</font>
<b>by </b><i>, <a class="ref" href="lfuzzy_0.html#T23">LFUZZY_0:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font>

;<br/>


</div><b>end;</b></div>
<a NAME="E4:44"/>
for <font color="Olive">m</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E5:44"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span>
 ;<br/>


</div>
