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<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_3.html#NM1" title="FUZZY_3: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"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></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> <font color="Maroon" title="c2">R</font></span>)</span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default">0 <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="c3">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><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></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/>.= 
<font color="Maroon" title="c3">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font>
<b>by </b><i><a class="ref" href="lfuzzy_1.html#T23" target="_self" title="LFUZZY_1:th.23">Th23</a></i>
;<br/>
<a NAME="E2:43"/><b>then </b><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:43"/><i><font color="Green" title="E25">A2</font></i>: 
for <font color="Olive" title="b1">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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>assume </b><a NAME="E1:43_2"/><i><font color="Green" title="E26">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <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="c3">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_1.html#T26" target="_self" title="LFUZZY_1:th.26">Th26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">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>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font>
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:43_2"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="lfuzzy_0.html#T23" title="LFUZZY_0:th.23">LFUZZY_0:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font>
<b>by </b><i><a class="ref" href="lfuzzy_1.html#T26" target="_self" title="LFUZZY_1:th.26">Th26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</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/>


</div><b>end;</b></div>
<a NAME="E4:43"/>
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#E2:43"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="lfuzzy_1.html#E3:43"><i><font color="Green" title="E25">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:43"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></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>
 ;<br/>


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