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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b><span class="p1">(<span class="default">$1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> $1 <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span>;<br/>
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font> = 
 <a href="fuzzy_4.html#K5" title="FUZZY_4:func.5">Imf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="lfuzzy_1.html#T24" target="_self" title="LFUZZY_1:th.24">Th24</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_1.html#T24" target="_self" title="LFUZZY_1:th.24">Th24</a></i>
;<br/>
<a NAME="E2:44"/><b>then </b><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E3:44"/><i><font color="Green" title="E25">A2</font></i>: 
for <font color="Olive" title="b1">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c5">m</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:44_2"/><i><font color="Green" title="E26">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span>
 ;<br/>

<i><font color="Green" title="E27">A4</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span>
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:44_2"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="lfuzzy_1.html#T27" target="_self" title="LFUZZY_1:th.27">Th27</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default">1 <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_1.html#T27" target="_self" title="LFUZZY_1:th.27">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_1.html#T25" target="_self" title="LFUZZY_1:th.25">Th25</a></i>
;<br/>
<b>hence </b><a NAME="E4:44_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E2:44_2"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:44"/>
for <font color="Olive" title="b1">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="lfuzzy_1.html#E2:44"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="lfuzzy_1.html#E3:44"><i><font color="Green" title="E25">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:44"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span>
 ;<br/>


</div>
