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<b>let </b><font color="Maroon">A</font> be  <a href="unialg_1.html#V6">partial</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_1.html#L1">UAStr</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">R</font> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">A</font> &amp; <font color="Maroon">R</font> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">R</font> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font> is <a href="relat_2.html#V3">symmetric</a> &amp; <font color="Maroon">R</font> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font> is <a href="relat_2.html#V8">transitive</a> );<br/>
<a NAME="E2:45"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, <a class="txt" href="pua2mss1.html#E1"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/><a NAME="E2:45_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="pua2mss1.html#K7" target="_self">|^</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="pua2mss1.html#K7" target="_self">|^</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="pua2mss1.html#E2"><i><font color="Green">Lemma25</font></i></a></i>;<br/><b>hence </b><a NAME="E3:45_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:45"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="txt" href="pua2mss1.html#E1:45"><i><font color="Green">E29</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:45"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">R</font> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Olive">i</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">R</font> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Olive">i</font> is <a href="relat_2.html#V3">symmetric</a> &amp; <font color="Maroon">R</font> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">A</font>,<font color="Olive">i</font> is <a href="relat_2.html#V8">transitive</a> )
 ;<br/>


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