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<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">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="E1:65"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">R</font> is <a href="lfuzzy_1.html#V3" target="_self">transitive</a> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:65"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">R</font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 
<b>by </b><i/>;<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><font color="Maroon">R</font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> ;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:65"/>
1 <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:65"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
;<br/>
<a NAME="E6:65"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">k</font> being non <a href="xboole_0.html#V1">empty</a>  <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="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:65_1"/>
<font color="Maroon">R</font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">R</font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:65_1"/><b>then </b>
<font color="Maroon">R</font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">R</font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:65_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:65"/>
for <font color="Olive">k</font> being non <a href="xboole_0.html#V1">empty</a>  <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="binarith.html#S1">BINARITH:sch 1</a>(, <a class="ref" href="lfuzzy_1.html#D1" target="_self">Def1</a>);<br/></i>
<a NAME="E8:65"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 
;<br/>
<b>hence </b><a NAME="E9:65"/>
<font color="Maroon">R</font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 ;<br/>


</div>
